# PAST PROBABILITY SEMINARS

## Fall, 2005

### Title and abstract

Wed. Oct.12, 3pm
Conference Room
Yevgeniy Kovchegov
Oregon State University
"On Mixing Times I"
Wed. Oct.19, 3pm
Covell 221
Ed Waymire
Oregon State University
"The anamolous diffusion problem for directed polymers"
Wed. Oct.26, 3pm
Covell 221
Robert Burton
Oregon State University
"Stationary Measures for Reinforcing Randomly Chosen Maps"
Tue. Nov.1, 1pm
Kidder 364
David Levin
University of Oregon
"A coupling, and conjectures of Darling and Erdos"
Wed. Nov.9, 3pm
Covell 221
Yevgeniy Kovchegov
Oregon State University
"On Mixing Times II"
Wed. Nov.16, 3pm
Covell 221
Ben Morris
University of California, Davis

"The spectral gap for the zero range process with constant rate"

Abstract. The zero range process with constant rate can be described as follows. Particles are distributed over the vertices of the d-dimensional torus. Each vertex has a clock that rings at rate one. When the clock of a (nonempty) vertex rings, a particle moves from that vertex to a randomly chosen neighbor. We obtain a tight bound for the spectral gap of this process, solving an open problem.
Wed. Nov.23
NO SEMINAR
NO SEMINAR NO SEMINAR THIS WEEK
Wed. Nov.30, 3pm
Covell 221
Ed Waymire
Oregon State University
"The anamolous diffusion problem for directed polymers II"

## Winter, 2006

The probability seminar will continue into the Winter quarter. We will meet on Wednesdays at 3pm in Kidder 280.
Here is a tentative list of speakers (to be extended): Mina Ossiander, Ed Waymire, Yevgeniy Kovchegov, Nathanael Berestycki (UBC), Jorge Ramirez, Robert T. Smythe (Department of Statistics)

### Title and abstract

Wed. Jan.11, 3pm
Kidder 280
Yevgeniy Kovchegov
Oregon State University

"Subcritical percolation: cluster expansion and Brownian bridge asymptotics"

Abstract. First,we will review basic facts about Bernoulli bond percolation model. Then,for a given point a in Z^d, we will show that a cluster in the d-dimensional subcritical Bernoulli bond percolation model conditioned on connecting points (0,...,0) and na if scaled by 1/(n||a||) along a and by 1/sqrt{n} in all orthogonal directions converges asymptotically to Time x (d-1)-dimensional Brownian bridge.
Wed. Jan.18, 3pm
Kidder 280
Yevgeniy Kovchegov
Oregon State University

"Critical percolation and Lorentz lattice gas model: an expository talk"

Abstract. Critical probability p_c=1/2 in 2D. Lorentz lattice gas model. Increasing events, pivotal edges and Russo's formula. Exponential decay in subcritical phase. Russo's formula adapted to LLG model.
Wed. Jan.25
NO SEMINAR
NO SEMINAR
NO SEMINAR THIS WEEK
Wed. Feb.1, 3pm
Kidder 280
Jorge Ramirez
Oregon State University

"Skew Brownian motion and diffusion: a model for heterogeneity"

Abstract. Solute transport in a medium with sharp discontinuities in the diffusion coefficient is studied via the identification of the associated stochastic process: skew diffusion. The model sheds light on the physical microscopic movement of solute particles in the presence of membranes, and connects back with the macroscopical PDE formulation via classic theory. Some interesting probabilistic properties and constructions of skew diffusion are discussed. The results are applied to a classic homogenization problem in layered medium.
Wed. Feb.8, 3pm
Kidder 280
Ed Waymire
Oregon State University

"An approach to unique invariant probabilities for Markov Processes"

Abstract. This talk is based on prior work with Rabi Bhattacharya and concerns conditions for existence of unique invariant probabilities for Markov processes on general state spaces in the absence of irreducibility.
Tue. Feb.14, 3pm
(Colloquium)
Dearborn 118
Nathanael Berestycki
University of British Columbia

" OF MICE AND MEN,... and random walks"

Abstract. We will see how tools from probability theory can help us answer some questions arising in the study of genome rearrangement, which have the following flavor: given two species (say, mice and men), can we quantify how different or how similar they are? On a mathematical level, this will lead us to study the behavior of a certain random walk on the symmetric group and show that it exhibits a phase transition. Along the way we will discuss some connections with Erdos-Renyi random graphs (aka mean-field percolation) and hyperbolic geometry. Some familiarity with elementary probability notions (such as the Poisson process) is preferable but not essential.
Wed. Feb.15, 3pm
BATCHELLER 250
Nathanael Berestycki
University of British Columbia

"Gibbs fragmentations"

Abstract. We study random partitions of 1,...,n where every cluster of size j can be in any of w(j) possible internal states. A Gibbs distribution is obtained by sampling uniformly among all possible configurations. Gibbs distributions arise naturally as equilibrium distributions of reversible coagulation - fragmentation processes. In this work we characterize irreversible processes where this microscopical equilibrium is moving towards a more fragmented state as time evolves. In particular we show that after reversing the direction of time they are the Marcus-Lushnikov coalescent process with affine collision rate K(x,y)=a+b(x+y) for some real numbers a and b. This is joint work with Jim Pitman (U.C. Berkeley).
Wed. Feb.22, 3pm
Kidder 280
Mina Ossiander
Oregon State University

"Random multiplicative cascade measures. Part I"

Abstract. The distribution of random cascade measures depends implicitly on that of an underlying cascade generator along with a branching number. This pair of talks will give some basic background and then discuss some interesting convergence and estimation issues. The material covered represents joint work with R. Keim, D. Rupp, and E. Waymire.
Wed. Mar.1, 3pm
Kidder 280
Mina Ossiander
Oregon State University

"Random multiplicative cascade measures. Part II"

Abstract. The distribution of random cascade measures depends implicitly on that of an underlying cascade generator along with a branching number. This pair of talks will give some basic background and then discuss some interesting convergence and estimation issues. The material covered represents joint work with R. Keim, D. Rupp, and E. Waymire.
Mon. Mar.6, 3pm
TBA
Jesus Rodriguez
North Carolina State University

"Recent Topics in Math Finance"

Abstract. With traditional financial derivatives having been so well studied, investment banks are looking for new areas to exploit market participants. This has led to a rush to study new areas. We will discuss possible directions for some of these areas. In particular we will discuss some probabilistic issues in credit derivatives and how ideas from energy derivatives can be used to tackle pricing issues in other areas.
Wed. Mar.15, 3pm
Kidder 280
Robert T. Smythe
Department of Statistics,
Oregon State University
"Yet Another Interval-Division Problem"
Abstract.

## Spring, 2006

The probability seminar will continue into the Spring quarter. We will meet on Wednesdays at 4pm in Kidder 364.
Here is a tentative list of speakers (to be extended): Larry Pierce, Anthony Quas (U. of Victoria), Weng-Keen Wong (Computer Science), David Levin (University of Oregon), Robert Burton, Yevgeniy Kovchegov

### Title and abstract

Tue. Apr.11, 3pm
(Colloquium)
Kidder 364
Krzysztof Burdzy
University of Washington

"On the Robin problem in fractal domains"

Abstract. The Robin boundary conditions represent the flow of a substance or heat through a semi-permeable membrane. Let u be a non-negative solution to the heat equation in a bounded domain with Robin boundary conditions. I will address the question of whether the infimum of u over the whole domain is equal to 0. I will stress the use of probabilistic techniques in the investigation of this purely analytic problem. The talk will be accessible to a broad mathematical audience and graduate students.

Joint work with R. Bass and Z.-Q. Chen.
Wed. Apr.19, 4pm
Kidder 364
POSTPONED
Larry Pierce
Oregon State University
POSTPONED
Wed. Apr.26, 4pm
Kidder 364
(Probability Seminar and Colloquium)
Anthony Quas
University of Victoria

"Maximal rates of divergence of ergodic averages along subsequences"

Abstract. Given a measure-preserving transformation T, there has been interest in the study of ergodic averages of the form
1/N[f(T^{a_1}x)+f(T^{a_2}x)+....+f(T^{a_N}x)].
For some sequences (a_n), these averages can be shown to converge pointwise for all measure preserving transformations, whereas for other sequences they diverge. In this talk, I'll describe the maximal rate of divergence and will apply the methods to review the negative solution to an old conjecture of Khinchine's (the original conjecture was as follows: given an integrable function g on the unit circle (written additively), the averages
1/N[g(x)+g(2x)+...+g(Nx)]
converge almost everywhere to the integral of g) (joint work with Mate Wierdl)
Wed. May 3, 4pm
Kidder 364
Larry Pierce
Oregon State University

"Computing entropy for Z^d-actions."

Abstract. We will explore new methods for the numerical approximation of entropy for Z^d-actions.
Wed. May 10, 4pm
Kidder 364
Stanley C. Williams
Utah State University

Abstract. (To be posted)
Wed. May 17, 4pm
NO SEMINAR
No seminar this week

Wed. May 24, 4pm
Kidder 364
Weng-Keen Wong
Computer Science, Oregon State University

"What's Strange About Recent Events: An Algorithm For the Early Detection of Disease Outbreaks"

Abstract. Syndromic surveillance is a new field with the goal of detecting disease outbreaks as early as possible using health-care data that precede diagnosis. Traditional outbreak detection algorithms detect disease outbreaks by looking for peaks in a univariate time series of health-care data. Current health-care surveillance data, however, are no longer simply univariate data streams. Instead, a wealth of spatial, temporal, demographic and symptomatic information is available. I will present a disease outbreak detection algorithm called What's Strange About Recent Events (WSARE), which uses a multivariate approach to improve detection time and accuracy. The algorithm itself incorporates a wide range of ideas, including association rules, Bayesian networks, hypothesis testing and permutation tests to produce a detection algorithm that is careful to evaluate the significance of the alarms that it raises.
Tue. May 30, 3pm
(Colloquium)
Kidder 364
Christopher Hoffman
University of Washington

"Uses of the Ergodic Theorem in Probability"

Abstract. The ergodic theorem is a powerful tool in probability. Its use involves considering a probabilistic model as a dynamical system. The ergodic theorem has been particularly useful in studying percolation, where it has been used to construct simple geometric proofs to a number of difficult problems. In this talk I will discuss three such applications: 1) Burton and Keane's proof that there is a unique infinite supercritical percolation cluster, 2) Berger and Biskup's proof that simple random walk on the infinite percolation cluster converges to Brownian motion, and 3) A proof that coexistence is possible in Richardson's growth model.
Wed. May 31, 4pm
Kidder 364
Christopher Hoffman
University of Washington
"Random Simplicial Complexes"

## Fall, 2006

The probability seminar will continue into the Fall quarter. We will meet on Thursdays at 2pm in Covell 221.
Here is a tentative list of speakers (to be extended): Stephen D. Scarborough, Robert Burton, Yevgeniy Kovchegov, Corina Constantinescu, David Levin (University of Oregon), Nick Meredith
Registration information: Math 607, CRN 16811

### Title and abstract

Thursday, Oct 5, 2pm
Covell 221
Yevgeniy Kovchegov
Oregon State University

"Mixing times via coupling method"

Abstract. An introduction into computing mixing times bounds and asymptotics with coupling techniques.
Thursday, Oct 12, 2pm
Covell 221
Yevgeniy Kovchegov
Oregon State University

"Mixing times via super-fast coupling"

Abstract. We provide a coupling proof that the transposition shuffle on a deck of n cards is mixing of rate $n\log(n)$ with a moderate constant. This has already been shown by Diaconis and Shahshahani but no natural coupling proof has been demonstrated to date. We also enlarge the methodology of coupling to include intuitive but nonadapted coupling rules, for example, to take in account future events and to prepare for their occurrence. (Joint work with R.Burton)
The paper for this talk can be found on Math ArXiv at http://front.math.ucdavis.edu/math.PR/0609568
Thursday, Oct 19, 2pm
Covell 221
Nick Meredith
Oregon State University

"Computing occupation times with integral equations"

Abstract. Consider a simple Markov process on [0,T]. The occupation times were extensively studied in the case of T increasing to infinity. Here we develop a method of computing the distribution for occupation times when T is small via integral equations and integral transforms.
Thursday, Oct 26, 2pm
Covell 221
Corina Constantinescu
Oregon State University

"Ordering of the Ruin Probabilities"

Abstract. In a renewal risk model the inter-claim times form a sequence of independent, identically distributed random variables. If the inter-claim times are distributed as a sum of n exponentials then a comparison between the ruin probabilities may be established for different n's. If, additionally, the insurance company invests in an asset with a price modeled by a geometric Brownian motion, a similar comparison between the ruin probabilities is presented. These comparisons are derived using sample path-wise domination and coupling arguments.
Thursday, Nov 2, 2pm
Covell 221
Robert Burton
Oregon State University

"Learning about Learning: using coupling paths to construct joint distributions for a pair of Markov chains"

Abstract. This should be an exposition that is fairly accessible. I will review coupling as a methodology for computing with Markov chains. This has had many payoff, from the understanding of mixing properties and then to the path Central Limit Theorems. It has been very useful in statistical physics and showing connectivity of random structures. It is also the natural tool for a looking at a Markov Chain with additional structure because they arise in a certain natural way: as the independent and identically distributed composition of random transformations. This object exploded fractals (and by association chaos theory) into the everyday realm of human consciousness.
Thursday, Nov 9, 2pm
Covell 221
Stephen D. Scarborough
Oregon State University

"Probability and Continued Fractions"

Abstract. This material was created for use as a possible project in a Senior Seminar. This talk will be very accessible for undergraduates. The required background is calculus and Laplace transforms. The needed number theory will be given in the talk.
Let x be a U(0,1) random variable. Express x as a continued fraction. x = [0;a(1),a(2),a(3),...]. We will find an expression for P(a(n)=k). Next we will closely examine P(a(2)=k). If time permits an expository discussion will be given for the asymptotic case.
Thursday, Nov 16, 2pm
Covell 221
David Levin
University of Oregon

"Glauber dynamics for Ising model on complete graph"

Abstract. This talk is mostly expository -- I describe the phase transition from fast to slow mixing for the Glauber dynamics for Ising on complete graph. The upper bound uses path coupling, and the lower bound uses Cheeger's inequality.
Thursday, Nov 30, 2pm
no seminar
NO SEMINAR

no seminar this week

## Winter, 2007

We meet on Thursdays at 2pm in Kidder 364.
Here is a tentative list of speakers: Mina Ossiander, Davar Khoshnevisan (University of Utah), Yan-Xia Ren (Peking University, visiting U.Oregon), Yevgeniy Kovchegov, Robert Burton
Registration information: Math 607, CRN 22637

### Title and abstract

Thursday, Jan 18, 2pm
Kidder 364
Mina Ossiander
Oregon State University

"Stein's Method: Part I"

Abstract. This is the first of a pair of seminars on Stein's method. It will be expository in nature. The plan is to give a sketch of Stein's original idea of characterizing distributions via certain expectation properties along with an application to normal approximation for sums of iid r.v.'s. Some refinements due to Chen and Shao will be included.
Thursday, Jan 25, 2pm
Kidder 364
Mina Ossiander
Oregon State University

"Stein's Method: Part II"

Abstract. This is the second of a pair of seminars on Stein's method. The plan is to indicate further applications of Stein's general approach to convergence of random variables. First we will see how his original idea can be used to characterize a variety of distributions. This will be followed by a new application of Stein's method to Polya's urn using a Stein characterization of the Beta distribution.
Thursday, Feb 1, 2pm
Joint with University of Oregon,
at University of Oregon,
Davar Khoshnevisan
University of Utah

"Potential theory for several Markov chains."

Abstract. In 1989, P. Fitzsimmons and T. Salisbury solved the long-standing open problem of describing exactly when the trajectories of two (or more) independent Markov processes intersect. Although the proof has been greatly simplified in the setting of Markov chains (Salisbury, 1996), this is still considered a very difficult result. In this talk, I will present a completely self-contained proof [still in the context of denumerable chains], which is based on arguments that were originally designed to study the Brownian sheet (Khoshnevisan and Shi, 1999).
Friday, Feb 2, 3pm
(Colloquium)
Kidder 364
Davar Khoshnevisan
University of Utah

"On the stochastic heat equation."

Abstract. Consider the classical heat equation in dimension d, and formally replace the external forcing term by white noise. The resulting "stochastic PDE" (SPDE, for short) is the so-called stochastic heat equation. It has been known for some time that the stochastic heat equation suffers from a "curse of dimensionality": It has function solutions if and only if the ambient dimension is one. First we present a rigorous formulation of this SPDE, and explain why it has function solutions in only one dimension. Then, we discuss some newly-found connections between systems of solutions and classical notions from geometric measure theory [joint work with Robert Dalang and Eulalia Nualart]. Time permitting, we also address the mentioned curse of dimensionality, in greater length, by presenting an unexpected connection to classical probabilistic potential theory and the theory of local times [joint work with Mohammud Foondun and Eulalia Nualart].
Thursday, Feb 8, 2pm
NO SEMINAR
No seminar this week

Thursday, Feb 15, 2pm
Kidder 364
Yan-Xia Ren
Peking University, visiting U.Oregon

"Limit theorems for super-diffusions corresponding to the operator Lu+βu-ku2 "

Abstract.

Thursday, Feb 22, 2pm
Kidder 364
Yevgeniy Kovchegov
Oregon State University

"On Stein-Chen coupling."

Abstract. This will be an expository talk on the Stein-Chen method. We will provide examples of applying Stein-Chen method, such as providing error estimates for Poisson approximation.
Thursday, March 1, 2pm
Kidder 364
Robert Burton
Oregon State University

"Stuck in the Middle Between Determinism and Randomness: The Birth of Chaos"

Abstract. This talk will be expository and will look at the origins of modern dynamical systems and the failure of the Newtonian program of determinism. This failure is implicit within Newtonian mechanics and does not rely on quantum effect. It turns out that under very general circumstances dynamical systems are observed as a finite events, for example, as on a computer monitor. This leads us to symbolic dynamics and formalism. From there Markov process arise naturally as do substitution like rules. These polar opposites are, in some sense, orthogonal. There will be no specific set of Differential Equations presented, rather we begin on one of the nicest and simplest objects, automorphisms of the torus.
Thursday, March 8, 2pm
NO SEMINAR
No seminar this week

Thursday, March 15, 2pm
NO SEMINAR
No seminar this week

## Spring, 2007

The probability seminar will continue into the Spring quarter. We will meet on Thursdays at 2pm in Kidder 364.
Here is a tentative list of speakers (to be extended): Jorge M. Ramirez, Tom Dietterich (School of Electrical Engineering and Computer Science), Serban Nacu (École Normale Supérieure, Paris), Ioana Dumitriu (University of Washington), Robert Burton, Ed Waymire, David Plaehn (InsightsNow, Inc.)
Registration information: Mth 507, Sec 003 - CRN 38288 and Mth 607, Sec 004 - CRN 38290

### Title and abstract

Wednesday, April 18, 12:00-12:50pm
(Eco-IGERT Colloquium)
Wilkinson 203
Serban Nacu
École Normale Supérieure, Paris

"Probability with Ants"

Abstract. Ants make an interesting subject for mathematicians. At the low level, an ant colony consists of a large number of similar individuals, performing fairly simple tasks, often with a random element.
Yet at the high level, the colony is capable of complex and robust behavior, achieved without central control. Information is transmitted across the colony by interactions among ants, and the behavior of individuals is strongly influenced by these interactions. Ants are also very successful creatures; it is estimated they make up at least 10% of the terrestrial animal biomass.
We discuss a series of experiments performed on a population of harvester ants living in the Arizona desert, that illustrate some of these points.
Their analysis raises some interesting statistical and mathematical questions. We also mention some problems in (pure) probability theory that were inspired by the ants. No previous knowledge of ants will be assumed.
This talk is based in part on joint work with Deborah Gordon and Susan Holmes
Thursday, April 19, 2pm
Kidder 364
Serban Nacu
École Normale Supérieure, Paris

"GENE EXPRESSION NETWORK ANALYSIS"

Abstract. There is a strong genetic component in cancer and many other diseases. Often the disease involves, or is caused by, dysfunction in the cellular machinery: genes in normal and disease cells behave in different ways. Those differences in gene expression can be measured using microarray technology, which in the recent past has become a fundamental technique in biology and medicine.
A typical microarray measures expression levels for 20,000 genes at the same time: this massive parallel power also raises important statistical and computational problems. After a brief introduction to the technology, we focus on the issue of gene interactions. Standard microarray analysis treats each gene separately, and ranks them according to some measure of differential expression. But in reality genes interact, they act in concert rather than alone; an analysis that accounts for those interactions has the potential to be more statistically accurate and biologically meaningful.
We introduce a method called GXNA (Gene eXpression Network Analysis). GXNA uses a gene interaction graph to search for clusters of related genes that are differentially expressed. It has several desirable features, such as fast runtimes and the computation of objective, permutation-based significance levels, and it shows promising results when applied to data sets involving cancer and the human immune system.
This is joint work with Rebecca Critchley-Thorne, Peter Lee, and Susan Holmes.
Friday, April 20, 3pm
(Colloquium)
Kidder 364
Serban Nacu
École Normale Supérieure, Paris

"Fast Coin Simulation"

Abstract. ou are given a coin with probability of heads p, where p is unknown. Can you use it to simulate a fair coin? How about a coin with probability of heads 2p? Or a coin with probability of heads f(p), where f is a known function? For a fair coin, the problem goes back to Von Neumann in the 1950s. In 1994, Keane and O'Brien obtained necessary and sufficient conditions for a function f to have such a simulation.
We are looking at the problem of efficient simulation. Let N be the number of p-coin tosses required to simulate a f(p)-coin toss. Typically N will be random; we say the simulation is fast if N is small, in the sense that its distribution has exponential tails. When does a function have a fast simulation?
Surprisingly, it turns out this happens if and only if the function is real analytic. This is an instance of a more general phenomenon: there is a connection between the computational and analytic/algebraic properties of certain systems. The proof is constructive, and leads to algorithms that can be implemented. We use tools from the theory of large deviations, approximation theory, and complex analysis.
This is joint work with Yuval Peres.
Thursday, April 26, 2pm
Kidder 364
Ioana Dumitriu
University of Washington

"Tridiagonal matrix models for general β ensembles"

Abstract. β-Hermite, -Laguerre, and -Jacobi ensembles are generalizations of the Gaussian, central Wishart, and MANOVA matrices, with applicability in fields like statistical mechanics and traffic pattern analysis.
Matrix models for these β-ensembles have been discovered relatively recently, and the implications of this discovery for the study of the β-ensemble eigenstatistics cannot be understated.
In this talk, we will show how these matrix models were discovered, and sketch the proof that the eigenvalues of the β-Hermite matrix have joint eigenvalue distribution given by the β-Hermite ensemble (which, for β=1,2, respectively 4, are the same as the eigenvalue distributions of the Gaussian Orthogonal, Unitary, respectively Symplectic ensembles).
Friday, April 27, 3pm
(Colloquium)
Kidder 364
Ioana Dumitriu
University of Washington

"Classical ensembles of random matrices: from the threefold way to a β future"

Abstract. In classical probability, the Gaussian, Chi-square, and Beta are three of the most studied distributions, with wide applicability. In the last century, matrix equivalents to these three distributions have emerged from nuclear physics (Gaussian ensembles) and multivariate statistics (Wishart and MANOVA ensembles). Their eigenvalue statistics have been studied in depth for three values of a parameter (β = 1, 2 and 4) which defines the "threefold way" and can be thought of as a counting tool for their real, complex, or quaternion entries.
The re-examination of the Selberg integral formula, in the late 80s, has brought the advent of general β-ensembles, which subsume the classical cases, and for which the Boltzmann parameter β acts as an inverse temperature. Their eigenvalue statistics interpolate between the isolated instances 1, 2, and 4, offering a "behind the scenes" perspective.
With the discovery of matrix models for the general β-ensembles in the early 00's, we have entered a new stage in the understanding of the complex phenomena that lie beneath the threefold way. While the β = 1,2 and 4 cases are and will always be special, we can now argue that the future of the classical ensembles is written in terms of a continuous β parameter.
Thursday, May 3, 2pm
Kidder 364
Robert Burton
Oregon State University

"Iterated Function Systems: The Fractal Nature of Discovery"

Abstract. Iterated Function Systems are a given by a composition of randomly chosen transformations of a separable metric space (ok, sometimes complete also). They are natural objects that arising in modeling, and as a coordinatization for Markov Chains with stationary transition probabilities. They have been going in and out of style, but they are guaranteed to raise a smile, in the sense that they have been often rediscovered in all innocence with new terminology. Here, in this part of the talk, I will insert a list of famous people over the last century and some friends (non-empty intersection) who have thought about these models.
Iterated Function Systems have seen a lot exposure lately because of Fractal Image Compression, using the idea that rough textures are often enough detail, and that this may be simulated with fast simple code. In fact, before they were used to produce detailed and attractive pictures, they were used as models of learning and adaptation. We give a very simple proof of the main idea, explore how they could model learning and give some results, with a hint at the methodology we used. If you just want the pictures go to www.electricsheep.org, itself a free open source downloadable software package and is an allusion to the greatest science fiction writer of all time.
Thursday, May 10, 2pm
Kidder 364
Tom Dietterich
School of Electrical Engineering and Computer Science,
Oregon State University

"Experience with Slice Sampling in the CALO Probabilistic Constraint Engine"

Abstract. This talk will present an overview of the CALO project and then discuss the Probabilistic Constraint Engine (PCE), which is the core inference engine in CALO. We have implemented a form of Slice Sampling (originally developed by Pedro Domingos at the University of Washington). This talk will describe the algorithm, our implementation, and initial experiments with it.
Thursday, May 17, 2pm
Kidder 364
Ed Waymire
Oregon State University

"Stochastic Particle Tracking and Skew Brownian Motion."

Abstract. In recent years a variety of stochastic particle tracking methods have been proposed, developed and compared for efficient numerical mass transfer computations in the engineering literature. Particle tracking microscopy is also an essential method of observation of heterogeneous protein diffusion in cellular biology. In this talk a particular stochastic particle method recently proposed and and shown to be superior in laboratory tests is explained as an unexpected byproduct of our earlier work on dispersion rates. This is based on a recent paper with B. Chastenet, J. Ramirez, E. Thomann, and B. Wood.
Thursday, May 24, 2pm
Kidder 364
Jorge M. Ramirez
Oregon State University

"Multiple Skew Brownian motion (MSBM) and applications"

Abstract. I construct MSBM as a generalization of skew Brownian motion to the case of infinitely many interfaces x_k, with k ranging over the integers. This process behaves like Brownian motion when away from the interfaces, and experiences a skewness (or localized drift) alpha_k at each x_k. The construction and most of the results are derived using the representation of MSBM as a scaling of Brownian motion under a random time change. Then the theory of Dirichlet forms is used to derive the L^2 semigroup of MSBM and connect it to a diffusion process with discontinuous coefficient. As an application, I give some results concerning advection-diffusion in a two dimensional layered medium, and an elementary proof of an arcsine law for skew Brownian motion.
Thursday, May 31, 2pm
Kidder 364
David Plaehn
InsightsNow, Inc.

"Mathematics in the Food Industry: Penalty Analysis"

Abstract. As an example of mathematics in the food industry, a method called (traditional) penalty analysis is considered. With penalty analysis, product developers try to better understand product deficiencies by looking at consumer responses to overall liking vs. so-called "just-about-right" (JAR) questions. JAR variables are categorical with response categories ranging from some degree of "too little" to 'just-about-right' to some degree of "too much". By calculating the overall liking for those consumers answering, "too little", for example, compared to those that answered 'just-about-right', one can assign a "penalty" for being "too little". It is shown that traditional penalty analysis is equivalent to ordinary least squares (OLS) regression, allowing for standard methods of significance testing of the penalties. A distribution for testing the so-called "penalty weights" is proposed and compared to a non-parametric method on a real data set.

## Fall, 2007

The probability seminar will continue into the Fall quarter. We will meet on Thursdays at 1:30pm in Covell 221.
Here is a tentative list of speakers (to be extended): Ed Waymire, Yevgeniy Kovchegov, Larry Marple (Electrical Engineering and Computer Science), Robert Burton, Yaroslav Bulatov (Electrical Engineering and Computer Science)
Registration information: Mth 507, Sec 003 - CRN 18859 and Mth 607, Sec 004 - CRN 15736

### Title and abstract

Thursday, October 4, 2pm
Covell 221
Ed Waymire
Oregon State University

"Size Biasing and Tree Polymer Models"

Abstract. An important size bias change of measure was introduced in the context of multiplicative cascades by Jacques Peyriere, also referred to as the "Peyriere probability", for the purpose of analyzing the structure of random measures obtained in the supercritical regime in which non-trivial cascade limits can be determined from the seminal 1976 paper of Kahane and Peyriere. Size biasing was extended by Stanley Williams and the author in 1994 as another approach to the Kahane-Peyriere existence theory for multiplicative cascades. Certain problems arising in the analysis of tree polymers will be shown to involve the analysis of cascade limits in the critical and subcritical cases, to which size biasing can effectively be applied. Some improvements on present theory are possible by this approach which will be presented.
Based on joint work with Stanley C Williams, and inspired by discussions with Harry Kesten during a sabbatical year at Cornell.
Thursday, October 11, 1:30pm
Covell 221
Yevgeniy Kovchegov
Oregon State University

"Occupation times and modified Bessel functions"

Abstract. For a given state of a continuous time Markov process, we use a combination of Fourier and Laplace transforms to express the distribution of the time spent by the process at that state within [0,t] time interval. Based on joint work with N.Meredith.
Thursday, October 18, 1:30pm
NO SENINAR
No seminar this week.

No seminar

Thursday, October 25, 1:30pm
Covell 221
Larry Marple
School of Electrical Engineering and Computer Science
Oregon State University

"DOUBLY TOEPLITZ COVARIANCE STRUCTURES FOR 2-D SIGNAL PROCESSING APPLICATIONS"

Abstract. Sophisticated multi-dimensional sensors are becoming more commonplace in radar, sonar, ultrasound medical imaging, and seismic signal collection systems. Statistical signal processing of acquired data from the sensors involves multi-dimensional and multi-channel (multivariate) covariance matrix structures for purposes of detection and target/feature classification. The size of matrices in actual sensor applications have dimensions of many thousands and require fast computational algorithms to make them feasible for real-time implementation. This presentation will show the some fast algorithm structures involving parametric means of estimating the covariances for the one-dimensional case to establish a baseline, and then will show some recent research efforts to develop the two-dimensional versions dealing with doubly Toeplitz (or Toeplitz-block-Toeplitz in some literature) structures.
Thursday, November 1, 1:30pm
Covell 221
Robert Burton
Oregon State University

"Shuffles, Couples, and Waffles"

Abstract. There has been interest in how long it should take a set, often a set with an algebraic structure, to uniformly mix. This time is infinite on arithmetic grounds. It is similar to asking how long it will take a binary expansion to equal 1/3. Instead, we back up a little and ask for mixing times. The amount of time (length of expansion) it takes to be closer than \epsilon to 1/3 is -log(\epsilong) where the log is to base 2. This rate is uniform over all points in the interval, not just 1/3. In our context, we ask how long will it take for mixing to be uniform with at least a fixed probability, say prob = 1/2. (This value is not usually important for the rate at which mixing occurs as function of the size of the set.) We may not mix exactly but we can get uniformly close. The method that is often used is coupling as a way to estimate the total variation norm. Often, this method cannot obtain the best mixing time. By 'tunnelling' into the future, we show how to improve the estimate on the time until it is optimal in the limit. We apply this to some simple models of shuffling card, and to a form of mixing, which we call matrix mixing.
Thursday, November 8, 1:30pm
NO SENINAR
No seminar this week.

No seminar

Thursday, November 15, 1:30pm
Covell 221
Yaroslav Bulatov
Oregon State University

"Belief Propagation in Graphical Models"

Abstract. Suppose we have a hidden Markov model with a sequence of states Y1,...,Yn and observations X1,...,Xn. An important quantity, for signal processing, and inference is the probability of some state Yi given all the observations X1,...,Xk. For HMM, this quantity can be found by marginalizing out the variables, which in signal processing community is known as the Forward-Backward algorithm. There's a generalization of the Forward-Backward algorithm to compute an analogous quantity for a wider class of probabilistic models, where interactions are described by a general tree graph, known as the Sum-Product algorithm, Message Passing or Belief Propagation. In addition to estimating the probability of a variable conditioned on the evidence, versions of this algorithm are used for decoding error-correcting codes, computing free energy in Ising-like models and providing heuristics for various NP-hard problems such as Min-Cut. In this talk I will describe the algorithm and some applications, some known results, and associated open problems.

## Winter, 2008

### Title and abstract

Thursday, January 24, 2:00pm
Kidder 364
Yevgeniy Kovchegov
Oregon State University

"Perfect coupling and tunneling to the future"

Abstract. We will compare coupling methods in probability to Perron-Frobeniuous decomposition. We will discuss perfect coupling and the method of "tunneling to the future" as a non-Markovian coupling technique that can be used to produce correct mixing rate.
Thursday, January 31
NO SEMINAR
No seminar today

Thursday, Feb 7, 2:00pm
Kidder 364
Robert Burton
Oregon State University

"Plausible Reconstruction of Mixtures of Genetic Information from the Sargasso Sea."

Abstract. We will talk about partial data from close to a decade's worth of data from a single location in the Sargasso Sea. Because of cost/time constraints, there is only partial information available. There are enzymes that cut the RNA at specific locations in specific finite patterns (like CCGG in the AGCT alphabet of nucleotides). We are then given estimates of the distance from an origin. Usually, one enzyme is used to estimate the percentage of certain bacteria. This experiment used three different enzymes in the hope of getting better estimates of the percentage of different organisms present, over months, depths, years.
We show partial progress in solving this model in a meaningful way.
There is a debate in the microbiology community as to whether the dominant method of evolution is from ancestor-derived genetic information or whether it is lateral organism-to-organism and organism-to-floating stuff. The answer is probably context and organism dependent. The difference is whether a tree model describes evolution (slow to adapt) or whether a network graph describes evolution (lots of variation and quicker to adapt). This data may show some surprising uniformity in the way that micro-organisms are constructed, giving support for the lateral method of adaptation.
Thursday, Feb 14, 2:00pm
Kidder 364
No seminar. The faculty will meet to discuss probability classes for next year.

No seminar. Probability offerings discussion.

Thursday, Feb 21, 2:00pm
Kidder 364
Raviv Raich
Electrical Engineering and Computer Science
Oregon State University

"Flow Cytometry: Manifold Learning of High Dimensional Data"

Abstract. The task of analyzing and processing high volumes of information poses a great challenge. We are interested in extracting a simple model that supports the complex data we observe to explain phenomena of interest. Geometry and more specifically manifolds offer means of explaining a low dimensional description of high dimensional data. One application of interest is Flow Cytometry, a technique that utilizes fluid dynamics to allow for individual identification of cells and statistical analysis of the sample as whole. In this presentation, we will demonstrate how learning Riemannian manifolds can be applied to Flow Cytometry for visualization, clustering, and classification of various cancer types.
Thursday, Feb 28, 2:00pm
Kidder 364
Torrey Johnson
Oregon State University

"Low-Dimensional Lattice Polymer Models."

Abstract. Sharp results are rare for low-dimensional lattice polymers. We will consider some examples of low-dimensional lattice polymer models for which such results are possible. These models examine the tendency of the polymer to localize near an "interface" (for us this will be the x-axis). We will also consider a mapping from the lattice polymer to the tree polymer which may be useful in obtaining results about the lattice polymer, which are usually more difficult to obtain.
Thursday, March 6, 2:00pm
CANCELED
Qi-Man Shao
University of Oregon
and
Hong Kong University of Science and Technology

"Stein's Method of Exchangeable Pairs with Application to the Curie-Weiss Model"

Abstract. PDF file
Thursday, March 13, 2:00pm
Kidder 364
Young Soo Seol
Oregon State University

"ON WEAK CONVERGENCE OF SKEW RANDOM WALK"

Abstract. The primary purpose of this presentation is to establish functional central limit theorem for skew random walks and to define skew Brownian motion as resulting weak limit. Since the skew random walk is just a symmetric random walk when away form the origin, the right scaling of space and time will be the same as in the functional central limit theorem for Brownian motion. Finally, we will show that skew Brownian motion is the weak limit of skew random walk proving the convergence of finite-dimentional distributions and tightness.

## Spring, 2008

### Title and abstract

Thursday, April 3, 2:00pm
Kidder 364
Herold Dehling
Ruhr-Universität Bochum, Germany

"The empirical pair process and a CLT for non-symmetric U-statistics."

Abstract. In my talk I will introduce and analyze the empirical pair process, and show how this concept can be applied to obtain a central limit theorem for non-symmetric U-statistics (joint work with Sergey Utev).
Thursday, April 10
NO SEMINAR
No seminar this week

Thursday, April 17, 2:00pm
Kidder 364
Ed Waymire
Oregon State University

"On the existence of infinite volume tree polymer limits"

Abstract. Existence of an infinite volume tree polymer model is discussed in the case of weak and strong disorder, and some results related to recent speculation by Yuval Peres and computer simulations by Torrey Johnson will be indicated -- leading to further speculations.
Thursday, April 24
NO SEMINAR
No seminar this week

Thursday, May 1, 2:00pm
Kidder 364
Robert Burton
Oregon State University

"Monotone Iterated Function Systems and Self-Reinforced Learning Models."

Abstract. The early years of the decade starting with 1950s saw the development of learning models which were Markov chains arising as iteration of random processes with monotone properties. There were papers by Bush and Mosteller, T.Harris, and S.Karlin who studied Markov Chains with the structure of iterated random composition of monotone functions on an interval. These ideas were predecessors to g-Martingales of M.Keane, and in turn led to Iterated Function Systems. We take this idea and generalize it to associative structure, i.e. those in which past repetitions of a behavior make this behavior more likely in the future. From this, we can get Central Limit Theorems, even in the absence of uniform mixing assumptions. This is seen as falling into the L_1 / L_2 dichotomy of analysis.
Thursday, May 8
NO SEMINAR
No seminar this week

Thursday, May 15, 2:00pm
Kidder 364
Qi-Man Shao
University of Oregon
and
Hong Kong University of Science and Technology

"Stein's Method of Exchangeable Pairs with Application to the Curie-Weiss Model"

Abstract. PDF file

## Fall, 2008

### Title and abstract

Thursday, October 9, 2:00pm
Kidder 350
Yevgeniy Kovchegov
Oregon State University

"Orthogonality and Probability"

Abstract. For a class of simple stochastic processes, we will compare the spectral approach of Karlin-McGregor to that of harmonic functions and martingales. We will explore one-dimensional models of random walks in random environment from the perspective of Riemann-Hilbert problems.
Thursday, October 16, 2:00pm
Kidder 350
No seminar this week
Thursday, October 23, 2:00pm
Kidder 350
Yevgeniy Kovchegov
Oregon State University

"Orthogonality and Probability, Part II"

Abstract. For a class of simple stochastic processes, we will compare the spectral approach of Karlin-McGregor to that of harmonic functions and martingales. We will explore applications of Karlin-McGregor diagonalization beyond nearst-neighbor models. Also, we will consider connection between one-dimensional models of random walks in random environments and the Riemann-Hilbert problems.
Thursday, October 30, 2:00pm
Kidder 350
Chris Orum
Oregon State University

"Fisher-Kolmogorov Equation"

Abstract. I will discuss the Fisher-Kolmogorov equation (a nonlinear, reaction-diffusion equation) and present a simple probabilistic argument using branching Brownian motion and Jensen's inequality that shows how its solutions may blow up in finite time.
Saturday, November 8
Building 99 at Microsoft
4820 NE 36th St, Redmond, WA 98052-5319
Northwest Probability Seminar

Northwest Probability Seminar

Thursday, November 13, 2:00pm
Kidder 350
Yevgeniy Kovchegov
Oregon State University

"Orthogonality and Probability, Part III: Beyond nearest neighbor transitions"

Abstract. The Karlin-McGreogor diagonalization can be used to answer recurrence/transience questions, as well as those of probability harmonic functions, occupation times and hitting times, and a large number of other quantities obtained by solving various recurrence relations, in the study of Markov chains. However with some exceptions (see Karlin-McGreogor 1975) those were nearest neighbor Markov chains on half-line. Grünbaum (2007) mentions two main drawbacks to the method as (a) «typically one cannot get either the polynomials or the measure explicitly», and (b) «the method is restricted to “nearest neighbour” transition probability chains that give rise to tridiagonal matrices and thus to orthogonal polynomials». In this talk we will give possible answers to the second question of Grünbaum for general reversible Markov chains. In addition, we will consider possible applications of the newer methods in orthogonal polynomials such as using Riemann-Hilbert approach, and their probabilistic interpretations.
Thursday, November 20, 2:00pm
Kidder 350
Thinh Nguyen
School of Electrical Engineering and Computer Science
Oregon State University

"Random Network Coding for Data Dissemination and Storage - Not Your Grandfather YouTube."

Abstract. Currently, to disseminate a piece of data from a source to multiple destinations on the Internet, the data is divided into a number of packets which are then forwarded hop-by-hop through a series of routers along the paths that connect the source and the destinations. This store-and-forward scheme is simple, however it is often suboptimal. In this talk, I will describe a new paradigm for data dissemination and storage over the Internet using Network Coding (NC) that promises significant performance improvements in both data dissemination throughput and data robustness, particularly in distributed settings. I will introduce some algebraic structures for a generic NC scheme then focus on a simple Random Network Coding scheme (RNC) that is immediately applicable to a variety of current and future Internet applications. I will describe the architecture and protocols of a distributed NC-based storage and delivery system that is superior to YouTube both in throughput and storage performance . A few theoretical results on the performance of RNC scheme will be given. These results are obtained using tools from basic probability, Markov chain, and birth-death process theories.

## Winter, 2009

### Title and abstract

Thursday, January 15, 2:00pm
Kidder 364
Chris Orum
Oregon State University

"Size-biased sum equation"

Abstract. I will show how the Mellin transform may be used to make progress on a probability problem that arises from Navier-Stokes equations.
Thursday, January 22, 2:00pm
Kidder 364
Yevgeniy Kovchegov
Oregon State University

"A note on adiabatic theorem for Markov chains and adiabatic quantum computation"

Abstract. We derive an adiabatic theorem for Markov chains using well known facts about mixing and relaxation times, and discuss the result in light of the recent advances in adiabatic quantum computation.
Thursday, January 29, 2:00pm
NO SEMINAR
No seminar this week

Thursday, February 5, 2:00pm
NO SEMINAR
No seminar this week

Thursday, February 12, 2:00pm
Kidder 364
Robert Burton
Oregon State University

"Some Combinatorial Models in Probability"

Abstract. The earliest forms of mathematical reasoning were probably geometry and combinatorics. Both areas are rich in accessible patterns. (Think of the geometric demonstration that the triangle numbers are n+1 choose 2' = n(n+1)/2.) We will show some simple patterns including tiling patterns, both random and deterministic. Examples are physical models such as Square Ice, Balanced Patterns, Substitution Schemes and other iterative models. Also, we will show methods (perhaps 'examples' is a better word here) of transforming patterns in hopes of better understanding and in search of patterns more amenable to rigorous thought and sound conclusions.
Thursday, February 26, 2:00pm
NO SEMINAR
No seminar this week

Thursday, March 5, 2:00pm
Kidder 364
Yevgeniy Kovchegov
Oregon State University

"Orthogonality and Probability, Part IV: Riemann-Hilbert problems, applications"

Abstract. The Karlin-McGreogor diagonalization can be used to answer recurrence/transience questions, as well as those of probability harmonic functions, occupation times and hitting times, and a large number of other quantities obtained by solving various recurrence relations, in the study of Markov chains. We will consider possible applications of the newer methods in orthogonal polynomials such as using Riemann-Hilbert approach, and their probabilistic interpretations.

## Spring, 2009

### Title and abstract

Tuesday, April 7, 4:00pm
Kidder 350
Yevgeniy Kovchegov
Oregon State University

"Mixing times via orthogonal polynomials"

Abstract. We will show examples of computing mixing times via orthogonal polynomials diagonalization of reversible Markov chains, the Karlin-McGregor approach. Also, we will discuss matrix Riemann-Hilbert problems in the context of reversible Markov chains.
Monday, April 13, 4:00pm
COLLOQUIUM,
Kidder 350
Jessica Zúñiga
Stanford University

"Merging of time inhomogeneous finite Markov chains"

Abstract. In this talk we will discuss the quantitative analysis concerning the asymptotic behavior of time inhomogeneous finite Markov chains. To study this behavior, we develop singular value techniques in the context of time inhomogeneous chains and introduce the notion of $c$-stability, which can be viewed as a generalization of the case when a time inhomogeneous chain admits an invariant measure. We describe some examples where these techniques yield quantitative results for time inhomogeneous chains. (This talk is on joint work with Laurent Saloff-Coste.)
Tuesday, April 21, 4:00pm
Kidder 350
Thilanka Appuhamillage
Oregon State University

"Trivariate density of skew Brownian motion, its local and occupation times and interesting distributions of skew Brownian motion"

Abstract. We compute the joint density of skew Brownian motion, its local time at the origin, and its occupation time of positive reals. In here we first introduce a notion of elastic skew Brownian motion and compute the corresponding Feynman-Kac formula. The result extends many of the formula known for Brownian motion due to Karatzas and Shreve, such as trivariate densities and arksine laws, to the case of skew Brownian motion. The problem was motivates by an application to advection-dispersion across an interface in heterogeneous porous media.
Tuesday, April 28, 4:00pm
Kidder 350
Postponed to May 26

Tuesday, May 12, 4:00pm
Kidder 350
Peter Otto
Willamette University

"Asymptotic behavior of the fluctuations of the magnetization near critical points"

Abstract. In this talk, I will present our recent work on the convergence rates of the magnetization and the corresponding fluctuations for a mean-field model in statistical mechanics. In particular, the proof of the existence of a critical speed to continuous phase transition points for the interaction and temperature parameters for which the magnetization falls within the fluctuations as their converge to zero.
Tuesday, May 19, 4:00pm
Kidder 350
No seminar: Lonseth Lecture at 3pm

No seminar: Lonseth Lecture at 3pm

Tuesday, May 26, 4:00pm
Kidder 350
Carlos Martins-Filho
Department of Economics
Oregon State University

"Nonparametric stochastic frontier estimation via profile likelihood"

Abstract. We consider the estimation of a nonparametric stochastic frontier model with composite error density which is known up to a finite parameter vector. Our primary interest is on the estimation of the parameter vector, as it provides the basis for estimation of firm specific (in)efficiency. Our frontier model is similar to that of Fan et al. (1996), but here we extend their work in that: a) we establish the asymptotic properties of their estimation procedure, and b) propose and establish the asymptotic properties of an alternative estimator based on the maximization of a conditional profile likelihood function. The estimator proposed in Fan et al. is asymptotically normally distributed but has bias which does not vanish as the sample size $n \rightarrow \infty$. In contrast, our proposed estimator is asymptotically normally distributed and correctly centered at the true value of the parameter vector. In addition, our estimator is shown to be efficient in a broad class of semiparametric estimators. A Monte Carlo study is performed to shed light on the finite sample properties of these competing estimators.

## Fall, 2009

### Title and abstract

Tuesday, October 6, 4:00pm
GILK 115
Yevgeniy Kovchegov
Oregon State University

"Intro to quantum probability and quantum computing I"

Abstract. In this expository talk we will begin with quantum gates and circuits, qubits, density matrix and quantum probability, covering the Grover's search algorithm in the end.
Tuesday, October 13, 4:00pm
GILK 115
Yevgeniy Kovchegov
Oregon State University

"Intro to quantum probability and quantum computing II"

Abstract. In this second talk we will concentrate on quantum Fourier transform and the phase estimation procedure.
Tuesday, October 20, 4:00pm
GILK 115
Yevgeniy Kovchegov
Oregon State University

"Intro to quantum probability and quantum computing III"

Abstract. In this third talk we will cover quantum Fourier transform, the phase estimation procedure and quantum walks.
Tuesday, October 27, 4:00pm
Postponed
Postponed

Tuesday, November 3, 4:00pm
GILK 115
Chris Sinclair
University of Oregon

"Random matrix theory"

Abstract. I will present a (very) brief introduction of quantities of interest in random matrix theory and why probabilists should care. I will then go on to explain a couple of ensembles I have worked on, and how I came to study random matrix theory via number theory.
Tuesday, November 10, 4:00pm
NO SEMINAR
No seminar this week

Tuesday, November 24, 4:00pm
GILK 115
Robert Burton
Oregon State University

"Uniformly Distributed Stationary Processes"

Abstract. (joint with Aimee Johnson) A sequences is uniformly distributed if for any interval J contained in the unit circle the proportion of points in the interval J of the first N points in the sequence tends to the length of J as N goes to infinity. One way to get such a sequence is to sample an iid sequence of uniform [0,1) random variables. The Central Limit Theorem implies that, with probability one, such a sequence is uniformly distributed with rate 1/n^{1/2}. One can ask for better rates than this but no better than 1/N because the error in a single point of the first N in a very small interval about the point is arbitrarily close to 1/N. Classically, people have looked at similar constructions involving powers of 1/K that arise from an ergodic process called the adding machine or else the K-adic process. Another construction, is for quadratic numbers ( or at least easy algebraic numbers) and uses the continued fraction expansion. Both of these are not stationary but arise from stationary processes that we generalize and compute the rate of uniform distribution. These both come as a kind of dual to iid processes or else Markov processes designed to maximize entropy. Then, at the end, we come back to highly random Markov processes by re-thinking and modifying the problem.
Thursday, December 3, 4:00pm
TBA
Mark Kelbert
Swansea University, UK

"Probabilistic representations for solutions of higher-order elliptic equations and polyharmonic functions"

Abstract. We study the so-called Lauricella problem for higher-order elliptic operators of the type $L=(\Delta+V)^m$. A Feyman-Kac type representation of the solutions is presented and the bounds for the growth of the solutions are proved by probabilistic methods.

## Winter, 2010

### Title and abstract

Tuesday, January 19, 4:00pm
WNGR 285
Yevgeniy Kovchegov
Oregon State University

"Discrete and continuous quantum walks"

Abstract. We will discuss the Hadamard, Grover and other types of discrete quantum walks, as well as continuous quantum walks. We will represent the one-dimensional Hadamard quantum walks via a classical Markov chain (joint work with R.Burton and T.Nguyen). Next, we will mention a unifying approach using quantum interchange walks (joint work with Z.Dimcovic).
Tuesday, January 26, 4:00pm
NO SEMINAR
No seminar this week
(but consider attending the statistics seminar on Monday, January 25, at 4pm in Kidder 364, where Dr. Robert Smythe will speak on "Collecting coupons: is there still some life in an old problem?" )

NO SEMINAR

Tuesday, February 2, 4:00pm
WNGR 285
Robert Burton
Oregon State University

"Square Ice, 3-Color Model, Hypercube Triangulations, Jones Knot Invariants, Exactly Solvable Models, Quantum Computation and their union into Temperley-Lieb Algebras"

Abstract. Since the title is almost longer than the talk, I will probably not completely cover everything promised. But this is a talk embedded in a series so things can always be pushed forward. The origin of this subject for me is my fascination with square ice. This is a 2-d model like everything in the talk in which every edge connecting 2-lattice points that have a distance 1 from each other has an arrow either up or down, or right or left. The configuration is complete and satisfies the condition that exactly 2 arrow heads touch a lattic point. This means there are '4 choose 2' = 6 local configurations possible at each point. This is known to be equivalent to the 3 color model in which the plane is tiled by 1 by 1 squares each colored one of Red, Blue, Green with the condition that no adjacent tiles have the same color. (There is also an equivalent dimer (i.e. domino) model. We show other equivalences. This model is connected with the modular group as can be seen by its connections with the Farey tree. Applying the formalism of the Farey tree to symbols instead of numbers gives a family of minimal Sturmian transformations and these code to create the matrices A_n, n = 1, 2, . . . so that (Trace((A_n)^n))^(1/n) converges to the entropy of these models. We can also look at the graph represented by these mostly 0-1 matrices and see that it is a kind of mediant triangulation of the hypercube. The normalized spectral radius of these graphs also give the entropy and other invariants of the square ice process. With a little squinting one can see the formalism of quantum mechanics within these models. Abstracting these models to Temperley-Lieb algebras leaves the beauty of these entropy relationships bare and connects them to the objects in the title and many more, including the quantum teaser.
Tuesday, February 16, 4:00pm
WNGR 285
NO SEMINAR: probability curriculum group meeting

NO SEMINAR: probability curriculum group meeting

Tuesday, February 23, 4:00pm
WNGR 285
Robert Smythe
Department of Statistics, Oregon State University

"Collecting coupons: is there still life in an old problem?"

Abstract. What do DeMoivre, Laplace, Euler, von Mises, Polya and Einstein have in common (besides being dead white guys)? All of them worked on some version of the Coupon CollectorÕs Problem, one of the oldest problems in combinatorial probability. After being worked over for three centuries by such eminent mathematicians, one might wonder if the problem has anything left to be done.
I will review some of the history of this problem, which in its classic form has to do with the number of ways to place k balls in n cells. Some recent results for a generalized version of this problem will also be described.
Tuesday, March 2, 4:00pm
WNGR 285
Dan Rockwell
Oregon State University

"What Makes Normal Numbers Normal and How Do We Find Them?"

Abstract. The idea of normal numbers dates back to 1909 in a paper by E. Borel. He proved that almost all real numbers are normal. So it would seem that finding a normal number would be easy. It turns out that it is not easy. I will explore the ideas of what numbers are known to be normal and the challenges of trying to prove a specific number is normal. I will also discuss a recent result by R. Isaac that may open the door to proving specific numbers are normal.
Tuesday, March 9, 4:00pm
WNGR 285
Thilanka Appuhamillage
Oregon State University

"Local occupation and first passage times of skew Brownian motion"

Abstract. The motivation for this work is the recently reported empirical observation of certain asymmetries in the breakthrough curves measured for a 1-D porous medium with the discontinuity in the dispersion coefficient perpendicular to the direction of the mean velocity. In this talk I will present some new results on the trivariate density of the position, local and occupation times of skew Brownian motion and their role in the solution of the breakthrough problem. Also I will present stochastic ordering of first passage times of skew Brownian motion as the explanation of the observed asymmetries depends on it. This talk is based on joint work with Vrushali Bokil, Enrique Thomann, Ed Waymire and Brian Wood (to appear).

## Spring, 2010

### Title and abstract

Tuesday, April 6, 4:00pm
GILK 104
Patricia Medina
Oregon State University

"The Dunford-Pettis Theorem, Orlicz Spaces and Related Topics"

Abstract. The goal of this talk is to make a link between uniformly integrable sets and relatively weakly compact sets. This will give a good motivation to introduce Orlicz Spaces, a generalization of L^p spaces.
Tuesday, April 13, 4:00pm
GILK 104
Hoe Woon Kim
Oregon State University

"Probabilistic representations of iterated Riesz transformations"

Abstract. Riesz transforms are a family of generalization of Hilbert transformation in one-dimensional space to n-dimensional space. They also are singular integral operators which are given by a convolution with the kernels having a singularity at the origin. At the seminar I will present probabilistic representations of Riesz transforms (Gundy-Varopoulos-Silverstein's background radiation and Bass' approach) and a new probabilistic representation of iterated Riesz transforms in the context of the incompressible Navier-Stokes equations.
Tuesday, April 20, 4:00pm
GILK 104
Patricia Medina
Oregon State University

"On the Convergence in Mean of Martingale Difference Sequences"

Abstract. This is the second part of a sequence of two talks. In our first talk, we gave two characterizations of uniform integrability. After that, we had a good setting to introduce Orlicz Spaces.
The aim of this second talk is to give probabilistic results involving Orlicz Spaces. First, we will talk about Freniche's results which states that weakly null martingale difference sequence in L^1[0,1] has arithmetic means that converge in norm to 0. We will focus on showing that any weakly null martingale difference sequence in an Orlicz space whose N-function belongs to complement class of {\Delta}_3 class has arithmetic means that converge in norm to 0.
Thursday, April 22, 4:00pm
GLSN 100
Anthony Quas
University of Victoria

"Lattices, Uniform Distribution and 3log(2)-pi^2/8"

Abstract. Starting from a very concrete dynamical problem involving piecewise isometries of the plane, we are led to elementary number-theoretic questions involving visible points of lattices. These yield surprising dynamical consequences.
Tuesday, May 4, 4:00pm
GILK 104
Oregon State University

"Adiabatic times for Markov chains and applications in statistical physics"

Abstract. The quantum adiabatic theorem is an important result in quantum mechanics. It states: A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum.
In this talk I apply the concepts of the quantum adiabatic theorem to a similar adiabatic theorem for discrete and continuous time Markov chains. I find general bounds on the 'adiabatic time' with respect to the mixing time and I apply this bound to Ising models with Glauber dynamics on different dimensional tori.
Outside of some basic probability and analysis knowledge, this talk is self-contained.

## Fall, 2010

### Title and abstract

Tuesday, October 19, 4:00pm
WNGR 201
Yevgeniy Kovchegov
Oregon State University

"Introduction to free probability"

Abstract. The topic of free probability is related to current research in random matrices, orthogonal polynomials and quantum probabilities. In this lecture I will cover the basic set-up of free probability theory.
Tuesday, November 16, 4:00pm
WNGR 201
Yevgeniy Kovchegov
Oregon State University

"Tokunaga self-similarity for time series"

Abstract. This talk is based on the joint work with Ilya Zaliapin of University of Nevada, Reno. Hierarchical branching organization is ubiquitous in Nature. It is readily seen in river basins, drainage networks, bronchial passages, cardiovascular systems, botanical trees, snowflakes, and lightning. Moreover, it can be used to describe the interaction of climate systems components, the fracture development in solids, structural organization of solid Earth, evolutionary relationships among biological species (phylogenetic trees), etc. Empirical evidence reveals a surprising similarity among various natural hierarchies - many of them are closely approximated by so-called self-similar trees (SST). This diverse class of trees is completely specified by a small number of parameters; this facilitates development of the SST theory and use of SSTs in applications. In this talk we show that the level set trees associated with certain types of time series and random models are Tokunaga SST.
Tuesday, November 23, 4:00pm
WNGR 201
Zlatko Dimcovic
Department of Physics
Oregon State University

"Discrete quantum walk on a binary tree"

Abstract. We have recently constructed a framework for quantum walks, based on classical walks with memory. This framework reproduces known walks, while it can be used to build walks in systems that are difficult for current approaches. As our first example of its utility, we study a symmetric discrete quantum walk on the infinite binary tree. For a walk starting from a pure state at a given level in the tree we compute the amplitude at the root, as a function of time and starting level. The result is strikingly different from the classical case, as its amplitude spans an order of magnitude, with a power law tail, while the classical one decays exponentially. (For example, for a delayed walk this property yields a polynomial vs. exponential speed up over the classical walk, in delay time.) The breadth of the probability peak indicates that any restriction of the extent of the tree, such as a matching tree, sinks or boundaries, would likely yield algorithms superior to classical. The calculation utilizes a variety of analytical techniques (memoried stochastic processes, combinatorics and path counting, transforms, steepest descent, orthogonal polynomials). This study also brings up interesting general questions about quantum processes on such structures. This talk is based on the joint work with Ian Milligan, Dan Rockwell, Robert M. Burton, Thinh Nguyen and Yevgeniy Kovchegov.
Tuesday, November 30, 4:00pm
WNGR 201
Mario Magaña
Electrical Engineering and Computer Science
Oregon State University

"Mitigation of Computer Platform Random Interference"

Abstract. The trend of electronic systems towards higher integration and higher performance has also brought new challenges to radio receiver design. The decrease of switching times accompanied by the increase of clocks and data rates, interconnection speeds contributes to improve the overall system performance. At the same time, they also affect wireless communications due to an increment of the emissions of electromagnetic radiation on the radio bands.
We have analyzed the main contributors to the noise generated by the emissions known as platform noise and some on their main features are presented.
Statistical analysis of platform noise measurements done on the 2.4 GHz band have shown that this type of noise is Non-Gaussian. Motivated by this fact, a statistical model that is consistent with the physical characteristics of the process rather than only on data fitting has been derived. The model is closely related to the one derived for modeling of clutter in radar and it has been found to agree with experimental data. Its analysis will be also presented. The accuracy of the noise model will allow us to design radios immune to it.
Finally, simulations of the impact on the radio performance of platform noise in terms of the Bit Error Rate (BER) for OFDM systems are presented.

## Winter, 2011

### Title and abstract

Thursday, January 13, 4:00pm
Kidder 350
David Levin
University of Oregon

"Random Walks on Combinatorial Designs"

Abstract. A combinatorial design is an array whose rows satisfy pairwise contraints, such as orthogonality. By studying an interesting family of random walks, we show that certain combinatorial designs exists, and develop asymptotic formulas for the number of such designs.

Joint work with W. de Launey.
Tuesday, February 22, 4:00pm
Bexell 323
Yevgeniy Kovchegov
Oregon State University

"Path coupling method"

Abstract. In this talk I will present the path coupling technique, and list applications in statistical mechanical models. Next, we will examine a coupling for random-to-random card shuffling process, and use a creative path coupling to find the bound on coupling time. This talk is based on joint work with Jennifer Thompson during Summer 2010 REU program.
Tuesday, March 1, 4:00pm
Bexell 323
Torrey Johnson
Oregon State University

"Some Results for Tree Polymers and Related Branching Random Walks"

Abstract. We consider a class of branching random walks that arise in connection with certain tree polymer models. A result characterizing a.s. connectivity of the support time-asymptotically is obtained for a range of parameter values. The mean of the (normalized) proportions of branching random walkers at integer sites {0,1,...,n} is exactly given by a binomial distribution. It is therefore natural to consider whether a CLT for the underlying random proportions might hold. Some connections with weak/strong disorder in the tree polymer will also be indicated. This is based on joint work with my PhD thesis advisor Ed Waymire.
Tuesday, March 8, 4:00pm
Bexell 323
Peter Otto
Willamette University

"Mixing times for the mean-field Blume-Capel model via aggregate path coupling"

Abstract. In this talk we investigate the relationship between the mixing times of the Glauber dynamics of a statistical mechanical system with its thermodynamic equilibrium structure. For this we consider the mean-field Blume-Capel model, one of the simplest statistical mechanical models that exhibits the following intricate phase transition structure: within a two dimensional parameter space there exists a curve at which the model undergoes a second-order, continuous phase transition, a curve where the model undergoes a first-order, discontinuous phase transition, and a tricritical point which separates the two curves. We determine the interface between the regions of slow and rapid mixing. In order to completely determine the region of rapid mixing, we employ a novel extension of the path coupling method, successfully proving rapid mixing even in the absence of contraction between neighboring states. This talk is based on joint work with Yevgeniy Kovchegov and Mathew Titus.

## Spring, 2011

### Title and abstract

Tuesday, May 3, 4:00pm
Kidder 364
Yevgeniy Kovchegov
Oregon State University

"Phase estimation in quantum computing"

Abstract. In this talk we will review one of the key techniques of quantum computation, the phase estimation. We will show how it is derived, and list the main applications.
Tuesday, May 10, 4:00pm
Kidder 364
Oregon State University

"Approximation of Rare Event Occurrence in Delay-Tolerant Networks with Poisson 'Clumping' "

Abstract. Motivated by a desire to study the performance limits of Delay-Tolerant Networks, we develop approximations for the occurrence times of (hopefully) rare events, using Poisson Clumping methods described in D. Aldous' 1987 book, Probability Approximations via the Poisson Clumping Heuristic. In a Delay-Tolerant Network, mobile nodes collect data continuously as they move around, and offload data when they reach an access point. But if they do not reach an access point in time, their tiny chip can run out of memory, risking data loss.

We demonstrate how to apply some basic theory from Brownian Motion and the Poisson Clumping Heuristic to determine the number of access points sufficient to minimize data loss. In particular, we derive a function that calculates bounds on the expected amount of time a mobile node spends without coverage, and functions that relate access point density to expected data loss and probability of overflow. Simulation results will also be presented that demonstrate the strengths and weaknesses of our approximations.
Tuesday, May 31, 4:00pm
Kidder 364
Nicholas Michalowski
Oregon State University

"From Orthogonal Polynomials to Random Walks"

Abstract. Recently, the orthogonal polynomial techniques of Karlin and McGregor were extended in order to determine the rates at which stationary reversible Markov processes converge to their stationary distribution. In the current work, we investigate the other direction by considering well-known sets of orthogonal polynomials and adjusting them by attaching a point mass at unity using the techniques of Koornwinder. These new orthogonal polynomials will represent a class of recurrent nearest-neighbor Markov chains over non-negative integers. The spectral properties of these Markov chains are studied. In the absence of spectral gap, we obtain polynomial rates of convergence to stationarity. This presentation is based on joint work with Y. Kovchegov.

## Fall, 2011

### Title and abstract

Thursday, October 20, 4:00pm
WNGR 149
Torrey Johnson
Oregon State University

"A central limit theorem for the stochastic Pascal's triangle and related results."

Abstract. I will define a type of branching random walk which we have termed the stochastic Pascal's triangle and present an a.s. central limit theorem for it, and a result characterizing its range. If time permits, some other results on tree polymers may also be discussed, as well as connections to weak and strong disorder for tree polymers. This is based on joint work with my thesis advisor, Edward Waymire.
Tuesday, October 25, 4:00pm
WNGR 201
Bartłomiej Siudeja
University of Oregon

"Heat kernel monotonicity for reflected Bessel processes."

Abstract. I will discuss monotonicity properties of Neumann heat kernels. I will prove that the probability of return to a starting point for the radial part of Brownian motion (Bessel process) is increasing when the point moves toward the boundary. This is plausible since the boundary reflects the process back toward the starting point. Surprisingly, this result holds only for dimensions 3 and higher. The proof involves "polar" random walk approximation for Brownian motion.
Tuesday, November 22, 4:00pm
WNGR 201
Oregon State University

"Adiabatic times for Markov chains and applications"

Abstract. In this talk I am going to define the adiabatic time for a time-inhomogeneous Markov chain. I am going to show how this time relates to an appropriate mixing time. I will expand the definition to briefly explore its application to different stochastic models. Finally, I will motivate the study of this topic with easy examples that are available to any student with a mild Linear Algebra background.
Tuesday, November 29, 4:00pm
WNGR 201
Robert Burton
Oregon State University

"The Poverty of Finite Words to the Riches of Dynamical Systems"

Abstract. The past years have seen the rise of various fields within mathematics. These are considered 'sexy' and have aesthetic components. These include the use of of Fractals, Chaos Theory, Complexity Theory, Formal Theories of Linguistics, and Symbolic Dynamical Systems. All of these have been birthed and raised within the Theory of Dynamical Systems.

This talk is expository, elementary, and designed to show how one generates dynamical systems from 'scratch.' A Dynamical System is a set X together with some mathematical structure such as a topology or a probability measure or some algebraic structure. Then we have a transformation T:X → X that preserves the mathematical structure on X. Here T represents the flow of time so that T('today') = 'tomorrow.' The question that we want to address is how do we take finite descriptions and turn them into infinite orbits within Dynamical Systems in an understandable way?

## Winter, 2012

### Title and abstract

Tuesday, February 7, 4:00pm
STAG 411
Yevgeniy Kovchegov
Oregon State University

Identification via completeness for discrete covariates and orthogonal polynomials

Abstract. We solve a class of identification problems for nonparametric and semiparametric models when the endogenous covariate is discrete with unbounded support. Then we proceed with an approach that resolves a polynomial basis problem for the above class of discrete distributions, and for the distributions given in the sufficient condition for completeness in Newey and Powell (2003). Thus, in addition to extending the set of econometric models for which nonparametric or semiparametric identification of structural functions is guaranteed to hold, our approach provides a natural way of estimating these functions. Finally, we extend our polynomial basis approach to Pearson-like and Ord-like families of distributions.

Based on joint work with N. Yildiz of University of Rochester.
Tuesday, February 21, 4:00pm
Kidder 350
Zachary Gelbaum
Oregon State University

"White Noise Representation of Gaussian Random Fields"

Abstract. We obtain a representation theorem for Banach space valued Gaussian random variables as integrals against a white noise. As a corollary we obtain necessary and sufficient conditions for the existence of a white noise representation for a Gaussian random field indexed by a compact measure space. As an application we show how existing theory for integration with respect to Gaussian processes indexed by [0,1] can be extended to Gaussian fields indexed by compact measure spaces.

## Spring, 2012

### Title and abstract

Tuesday, April 24, 4:00pm
Kidder 364
Yevgeniy Kovchegov
Oregon State University

"Application of aggregate path coupling and large deviations to mixing times of statistical mechanical models"

Abstract. We investigate the relationship between the mixing times of the Glauber dynamics of a statistical mechanical system with its thermodynamic equilibrium structure. For this we consider the mean-field Blume-Capel model (a.k.a. Blume-Emery-Griffiths model), one of the simplest statistical mechanical models that exhibits the following intricate phase transition structure: within a two dimensional parameter space there exists a curve at which the model undergoes a second-order, continuous phase transition, a curve where the model undergoes a first-order, discontinuous phase transition, and a tricritical point which separates the two curves. We determine the interface between the regions of slow and rapid mixing. In order to completely determine the region of rapid mixing, we employ a novel extension of the path coupling method, successfully proving rapid mixing even in the absence of contraction between neighboring states.

Next, I will briefly explain generalizing the method to a class of statistical mechanical models satisfying large deviation principle, such as the mean-field Potts model. Based on the joint work with Peter Otto of Willamette University.
Friday, May 11, 4:00pm
CANCELED
F. Alberto Grünbaum
UC Berkeley

"Spectral Methods for Discrete Time Quantum Walks, limit times, recurrence"

Abstract. I will show how to associate a discrete time quantum walk to any CMV matrix, use this to compute a few limit laws (the analogs of the Gaussian in the classical case) and then introduce a definition of recurrence for QWs that is different from the one introduced in 2008 by Stefanak, Jex and Kiss. I will try to make the point that many classical tools of analysis from the 1920's are useful in the study of QWs and that 'in turn this relatively new field gives new open problems in a very well established area of analysis. Indeed QWs give a way to reopen problems considered by J. Wallis and L. Euler around 1700.
Tuesday, June 5, 4:00pm
Kidder 364
Oregon State University

"Stable adiabatic times for Markov chains"

Abstract. In this talk I will define a time-inhomogeneous Markov chain and a measurement to determine its stability called the stable adiabatic time and I will provide a bound for this measurements in terms of a mixing time. There is a quantum analogue of this process, and we compare the bound in our result to a bound of the quantum adiabatic time. Outside of a basic understanding of Markov chains, this talk is self-contained.

## Fall, 2012

### Title and abstract

Tuesday, October 23, 3:00pm
Kidder 278
Zachary Gelbaum
Oregon State University

"Fractional Brownian Fields Over Manifolds"

Abstract. Brownian motion is the fundamental example of a stochastic process, that is, a real valued random function over the real line; what is its analogue if we replace the real line with some other manifold? Does any analogue exist? These questions were first considered by Paul Levy around 1950 for the case of Euclidean space and the sphere. Since then a number of researchers have looked at this question, considering also fractional Brownian motion, however until recently satisfactory extensions only existed for some special classes of manifolds and only certain fractional brownian motions. In this talk we will discuss a new approach whereby analogues of the full range of fractional Brownian motions are constructed over a wide class of manifolds. In doing so we will highlight a connection between fractional Brownian motion and the Laplacian that doesn't seem widely known (although we think it should be!). The tools we use come from spectral geometry, in particular we make essential use of the heat kernel and its estimates.
Tuesday, November 20, 3:00pm
Kidder 278
Son Luu Nguyen
Oregon State University

"Pathwise Convergence Rate for Numerical Solutions of Stochastic Differential Equations" (joint work with Prof. George Yin)

Abstract. Devoted to numerical solutions of stochastic differential equations (SDEs), in this talk we construct a sequence of re-embedded numerical solutions having the same distribution as that of the original SDE in a new probability space. It is shown that the re-embedded numerical solutions converge strongly to the solution of the SDE. Moreover, the rate of convergence is ascertained. Different from the well-known results in numerical solutions of SDEs, in lieu of the usually used Brownian motion increments in the algorithm, an easily implementable sequence of independent and identically distributed (i.i.d.) random variables is used. Being easier to implement compared to the construction of Brownian increments, such an i.i.d. sequence is preferable in the actual computation. As far as the convergence and uniform mean squares error estimates are concerned, the use of the i.i.d. sequence does not introduce essential difficulties compared with that of the Brownian increments. Nevertheless, the analysis becomes much more difficult for the study of rates of convergence because one has to deal with the diference of the Brownian increments and the i.i.d. sequence in the almost sure sense. Our work presents a new angle of ascertaining the convergence rates.
Tuesday, November 27, 3:00pm
Kidder 278
Thinh Nguyen
EECS, Oregon State University

"Design Robust Network Protocols For Non-Stationary Environment via Fast Mixing Times"

Abstract. We present a new analytical framework for designing network protocols and application policies under non-stationary environments. In order to cope with non-stationary environments, e.g., fluctuating nature of traffic volume or fading in wireless channels, it is preferable to have a network protocol that adapts and quickly achieves a given objective. In this context, we use the concept of mixing time in Markov chain theory to study the convergence rates of different protocol designs. As an application of our framework, we propose a convex optimization approach for obtaining the robust queuing policy that drives any initial distribution of packets in the queue to the target distribution in the fastest time. We then augment the proposed framework to obtain a queuing policy that produces an epsilon-approximation to the target distribution with even faster convergence time. The augmented framework is useful in dynamic settings where the traffic statistics changes frequently, and thus fast adaptation is preferable. Both simulation and theoretical results are provided to verify our approach.

## Winter, 2013

### Title and abstract

Tuesday, February 5, 3:00pm
BEXL 322
Jeff Steif
Chalmers University of Technology, Sweeden

"The many faces of the T T-inverse process"

Abstract. The T T-inverse process or equivalently "random walk in random scenery" is a family of stationary processes that exhibits an amazing amount of behavior. Each random walk yields such a process and as you vary the behavior of the random walk, you obtain essentially all possible ergodic theoretic behaviors. There is also a phase transition that arises which we can only partially prove. This work is done jointly with a number of people including Frank den Hollander, Mike Keane and Sebastien Blachere.
Tuesday, February 19, 3:00pm
BEXL 322
Yevgeniy Kovchegov
Oregon State University

Horton self-similarity of Kingman's coalescent tree

Abstract. We establish a weak form of Horton self-similarity for tree representation of Kingman's coalescent process and, equivalently, level-set tree of a white noise. The proof is based on a Smoluchowski-type system of ordinary differential equations for the number of Horton-Strahler branches in a tree that represents Kingman's coalescent via a hydrodynamic limit. We conjecture, based on numerical observations, that the Kingman's coalescent is also Horton self-similar in regular strong sense with Horton exponent 0.328533... and asymptotically Tokunaga self-similar. Finally, we demonstrate combinatorial equivalence between the trees of a Kingman's coalescent and level-set trees of a discrete white noise. This talk is based on joint work with Ilya Zaliapin of University of Nevada.

## Spring, 2013

### Title and abstract

Tuesday, April 23, 3:00pm
BAT 250
Partha Dey
Courant Institute of Mathematical Sciences
New York University

"Smart path technique for Gaussian comparison"

Abstract. Smart path technique is a powerful method for comparing two Gaussian processes with different covariance structure. In this talk I will explain the technique with examples from spin glasses, proving hypercontractivity for Gaussian and proving extremal point process convergence for correlated Gaussians.
Tuesday, June 4, 3:00pm
BAT 250
Edward C. Waymire
Oregon State University

"A Representation of the Tree Polymer Partition Function under Strong Disorder"

Abstract. A formula for the asymptotic distribution of the partition function of random energy cascades associated with tree polymers under strong disorder is obtained. The approach also provides a new representation of Aidekon's constant C* for the asymptotic distribution of extremes of a branching random walk. This is based on joint work with Partha Dey, Courant Institute for Mathematical Sciences.

## Fall, 2013

### Title and abstract

Tuesday, November 26, 3:00pm
GILK 104
Yevgeniy Kovchegov
Oregon State University

"On Markov Chain Monte Carlo"

Abstract. This is an expository talk on Markov Chain Monte Carlo (MCMC) sampling techniques. We will show how sampling with MCMC works and how to solve optimization problems with MCMC, and tune MCMC with simulated annealing. We will discuss how to use MCMC for inference in Baysian networks, and state of the art methods for estimating the running times of MCMC algorithms.
Tuesday, December 3, 4:00pm
Kidder 278
Peter T. Otto
Willamette University

"Rapid mixing of Glauber dynamics of Gibbs ensembles via aggregate path coupling and large deviations methods"

Abstract. We present a novel extension to the classical path coupling method to statistical mechanical models which we refer to as aggregate path coupling. In conjunction with large deviations estimates, we use this aggregate path coupling method to prove rapid mixing of Glauber dynamics for a large class of statistical mechanical models, including models that exhibit discontinuous phase transitions which have traditionally been more difficult to analyze rigorously. While here we apply the aggregate path coupling method to statistical mechanical models, the general methodology can be applied to other relevant Markov chains.

## Winter, 2014

### Title and abstract

Tuesday, January 14, 3:00pm
FURM 105
Andrey Sarantsev
University of Washington

"Exponential convergence for the gap process of rank-based competing Brownian particles"

Abstract. Consider a finite system of competing Brownian particles on the real line. Each particle moves as a Brownian motion with drift and diffusion coefficients, depending on its rank. Consider the gaps between consecutive particles; under some conditions, it converges exponentially fast to a certain stationary distribution. The question is, how exactly fast? We find estimates of the exponent of convergence. Our main tool is Foster-Lyapunov functions theory; we also apply stochastic comparison techniques.
Tuesday, January 21, 3:00pm
FURM 105
Albert M. Fisher
University of São Paulo

"Frobenius theorems for matrix sequences"

Abstract. For a single nonnegative dxd matrix, there are these three classical results: the Perron-Frobenius Theorem (a primitive matrix has a unique nonnegative eigenvector), the Frobenius decomposition theorem (the elements 1,2,...,d can be grouped into communicating states, equivalently the matrix has an upper triangular block form, with primitive or zero blocks on the diagonal after taking some power) and the Frobenius-Victory theorem (which identifies eigenvectors of such an upper triangular matrix in terms of the primitive blocks and "distinguished eigenvalues".)
In this talk we describe nonstationary versions of these theorems, which are key tools in a study of the ergodic theory of adic transformation.s (Joint with Marina Talet of Aix-Marseille University).
Thursday, February 13, 12:00pm
(Mathematical Biology Seminar)
Kidder 236
Yevgeniy Kovchegov
Oregon State University

"Noise in gene regulatory networks, FKG and Holley inequalities"

Abstract. Gene regulatory networks are commonly used for modeling biological processes and revealing underlying molecular mechanisms. The reconstruction of gene regulatory networks from observational data is a challenging task, especially, considering the large number of involved players (e.g. genes) and much fewer biological replicates available for analysis. We propose a new statistical method of estimating the number of erroneous edges that strongly enhances the commonly used inference approaches. This method is based on special relationship between correlation and causality, and allows to identify and to remove approximately half of erroneous edges. Using positive correlation inequalities we established a mathematical foundation for our method. Joint work with A. Yambartsev, M. Perlin, N. Shulzhenko, K.L. Mine, X. Dong, and A. Morgun.
Tuesday, February 25, 3:00pm
FURM 105
David Koslicki
Oregon State University

"Substitution Markov Chains and Martin Boundaries"

Abstract. Substitution Markov chains (SMC's) are a kind of randomization of deterministic substitution dynamical systems and were recently introduced as a new model to describe molecular evolution. Here I will show some additional basic properties of this class of countable state Markov chains and give an example for which the Martin boundary can be explicitly calculated and is found to be the product space of the unit interval and the full 2-Bernoulli shift.
Tuesday, March 4, 3:00pm
FURM 105
Bartłomiej Siudeja
University of Oregon

"The hot spots conjecture for acute triangles"

Abstract. I will describe the hot spots conjecture of J. Rauch, known results (probabilistic and analytic) and counterexamples. After that I will consider arguably the simplest unknown case: acute triangles. I will discuss recent progress sparked by a new method due to Miyamoto.

## Spring, 2014

### Title and abstract

Tuesday, April 29, 4:00pm
Kidder 350
Benjamin Young
University of Oregon

"Inverting the Kasteleyn matrix for the Aztec Diamond"

Abstract. I'll describe how to invert the Kasteleyn matrix for a few of the most interesting weight functions on the Aztec Diamond. This, in principle, allows the computations of correlation kernels for the associated tiling model. In some of the simpler cases, it is possible to compute asymptotics for this kernel rigorously; though this is essentially a combinatorics talk, I'll mention the sorts of asymptotics which can be derived. In more difficult cases (specifically 2-periodic weights) the asymptotics are not yet tractable, although they represent perhaps the most promising way in which one might study the transition between liquid and gaseous regimes in a dimer model. Joint work with Sunil Chhita and Kurt Johansson.
Tuesday, May 6, 4:00pm
Kidder 350
Zhen-Qing Chen
University of Washington

"Anomalous diffusions and fractional order differential equations"

Abstract. Anomalous diffusion phenomenon has been observed in many natural systems, from the signalling of biological cells, to the foraging behaviour of animals, to the travel times of contaminants in groundwater. I will first discuss the connections between anomalous diffusions and differential equations of fractional order, and then present some recent results in the study of heat kernels for non-local operators of fractional order.
Tuesday, May 13, 4:00pm
Kidder 350
Yung-Pin Chen
Lewis & Clark College

"Making two integers coprime more likely"

Abstract. I will introduce a Markov chain on the set of positive integers with the following transition probabilities: An integer will visit equally likely those integers that are coprime to it. I will discuss the probability of selecting two coprime integers if they are generated from the stationary distribution of this Markov chain. I will also discuss the evaluation of some series involving the Euler totient function.
Tuesday, June 3, 4:00pm
Kidder 350
Sooie-Hoe Loke
Oregon State University

"On the Hitting Times of Integral of Geometric Brownian Motion"

Abstract. The connection between Bessel process and the integral of geometric Brownian motion (IGBM) has been well-established. The key to this approach is the Lamperti relation. However, a common difficulty is that arguments constructed for Bessel processes with positive index generally do not carry over to the ones with negative index. In this talk, we use a differential equation approach to study the hitting times of IGBM. We discuss the paper by Metzler (2013) in which the Laplace transform of hitting times is expressed in terms of the gamma and confluent hypergeometric functions. The transform satisfies Kummer's equation which is obtained using Ito's formula and standard results on hitting times of diffusion processes.

## Fall, 2014

### Title and abstract

Tuesday, October 21, 4:00pm
Kidder 238
Anatoly Yambartsev
University of São Paulo, Brazil

"Large deviations for excursions of M/M/∞"

Abstract. We derive a large deviations principle for the trajectories generated by a class of ergodic Markov processes. Specifically, we work with M/M/∞ queueing processes. We study large deviations of these processes scaled equally in both space and time directions. Our main result is that the probabilities of long excursions originating at state 0 would converge to zero function with the rate proportional to the square of the scaling parameter. The rate function is expressed as an integral of a linear combination of trajectories.
Tuesday, November 11, 4:00pm
Kidder 238
Debashis Mondal
Department of Statistics, Oregon State University

"Applying Dynkin's isomorphism: an alternative approach to understand the Markov property of the de Wijs process"

Abstract. Dynkin's (1980) seminal work associates a multidimensional Markov process with a multidimensional Gaussian random field. This association, known as Dynkin's isomorphism, has profoundly influenced the studies of Markov properties of generalized Gaussian random fields. In this talk, applying Dykin's isomorphism, we shall investigate a particular generalized Gaussian Markov random field, namely, the de Wijs process that originated in Georges Matheron's pioneering work on mining geostatistics and, following McCullagh (2002), is now receiving renewed attention in spatial statistics. Dynkin's theory grants us further insight into Matheron's kriging formula for the de Wijs process and highlight previously unexplored relationships of the central Markov models in spatial statistics with random walks and the Brownian motion on the plane.
Tuesday, November 25, 4:10pm
Kidder 356
Anatoly Yambartsev
University of São Paulo, Brazil

"Phase transition in ferromagnetic Ising model with a cell-board external field"

Abstract. We show the presence of a first-order phase transition for a ferromagnetic Ising model on integer 2 dimensional lattice with a periodical external magnetic field. The external field takes two values h and -h, where h>0. The sites associated with positive and negative values of external field form a chessboard configuration with rectangular cells of sides L_1xL_2 sites. The phase transition holds if h is small enough. We prove a first-order phase transition using reflection positivity (RP) method. We apply a key inequality which is usually referred to as the chessboard estimate. This is a joint work with E. Pechersky and M. Gonzalez, my PhD student.

## Winter, 2015

### Title and abstract

Thursday, January 29, 12:00pm
WNGR 201
Jesus Perez
University of Cantabria, Spain

"Dynamic resources allocation in wireless links. A MDP formulation."

Abstract. This work addresses the problem of adaptive modulation and power in wireless systems with a strict delay constraint. Modulation and transmit power are dynamically adapted to minimize the outage probability for a fixed data rate. A discrete-time stationary Markov chain is used to model the time-varying channel. The problem is formulated as a finite-horizon MDP. The solution is a set of power/modulation allocation policies to be used during the transmission, as a function of the channel and system state. Numerical results show the benefits of such adaptation policies.
Thursday, February 19, 12:00pm
WNGR 201
Hyungho Youn
Seoul Institute, Seoul, Korea

"A Dynamic Cournot Model with Brownian Motion"

Abstract. In this talk we consider a stochastic version of a dynamic Cournot model. The model is dynamic because firms are slow to adjust output in response to changes in their economic environment. The model is stochastic because management may make errors in identifying the best course of action in a dynamic setting. We capture these behavioral errors with Brownian motion. The model demonstrates that the limiting output level of the game is a random variable, rather than a constant that is found in the non-stochastic case. In addition, the limiting variance in firm output is smaller with more firms. Finally, the model predicts that firm failure is more likely in smaller markets and for firms that are smaller and less efficient at managing errors. This talk is based on joint work with Victor J. Tremblay that was published in Theoretical Economics Letters.

## Spring, 2015

### Title and abstract

Thursday, April 16, 12:00pm
GILK 115
Matthew Junge
University of Washington

"The frog model on trees."

Abstract. On a d-ary tree place some number (random or otherwise) of sleeping frogs at each site, as well as one awake frog at the root. Awake frogs perform simple random walk and wake any "sleepers" they encounter. A longstanding open problem: Does every frog wake up? It turns out this depends on d and the amount of frogs. The proof uses two different recursions and two different versions of stochastic domination. Joint with Christopher Hoffman and Tobias Johnson.
Thursday, April 30, 12:00pm
Furman 202
Joint with Math Bio Seminar
(job talk)
Benjamin Dalziel
Department of Ecology and Evolutionary Biology
Princeton University

"Deterministic chaos in US measles epidemics."

Abstract. Regular fluctuations in the incidence of immunizing infections exemplify the emergence of stable patterns in complex populations. Stable annual or biennial cycles in disease incidence occur in a variety of host-pathogen systems because of a common demographic clockwork, consisting of depletion of the susceptible population by infection or vaccination followed by recruitment through birth or waning immunity, modulated by seasonal fluctuations in transmission rates. High amplitude fluctuations in transmission rates can cause nonlinear dissonances in the demographic clock, leading to unpredictable variation in epidemic sizes that show sensitive dependence on initial conditions/deterministic chaos. However, this is hypothesized to be rare, because sufficiently large oscillations in transmission rates are uncommon, and would result in deep epidemic troughs that predispose the system to stochastic extinction. I will discuss recent work with collaborators analyzing epidemic data that describe a ubiquitous path to locally persistent deterministic chaos through small shifts in the seasonal pattern of transmission, rather than through high amplitude fluctuations in transmission rates. We base our analysis on a comparison of measles incidence in 80 major cities in the prevaccination era US and UK. Unlike the regular limit cycles of the UK series, the US data exhibit spontaneous shifts in epidemic periodicity, due to a slight lengthening of the seasonal period of low transmission associated with school summer holidays. These local dynamics resulted in spatially decorrelated epidemics across the US during the early 20th century. This shows that subtle systematic changes in host behavior can fundamentally alter the spatiotemporal coherence of epidemics, without significantly impacting pathogen persistence, globally or locally. Routes to deterministic chaos in population dynamics may therefore be prevalent.
Thursday, May 14, 12:00pm
GILK 115
Peter T. Otto
Willamette University

"Expected length of random minimum spanning trees."

Abstract. Consider a graph where each edge is given an independent uniform [0,1] length. In 1985, Frieze proved that the expected length of the minimum spanning tree with these random edge lengths of the complete graph converge as the number of vertices go to infinity. Since then there have been numerous refinements and generalizations of this result. In this talk, I will give a survey of some of these results including the work we completed during the Willamette Valley REU Consortium for Mathematics Research in 2008 where we derived a polynomial representation of the expected length of the minimum spanning tree.
Thursday, June 4, 12:00pm
GILK 115
Ilya Zaliapin

"Horton and Tokunaga self-similarity for random trees: Empirical evidence and rigorous results"

Abstract. Nature exhibits many branching tree-like structures beyond the botanical trees. River networks, Martian drainage basins, veins of botanical leaves, lung and blood systems, and lightning can all be represented as tree graphs. In addition, time-oriented trees describe a number of dynamic processes like spread of disease or transfer of gene characteristics. This would sound like a trivial observation if not for the following fact. Despite their apparent diversity, a majority of rigorously studied branching structures exhibit simple two-parametric Tokunaga self-similarity and Horton scaling. The Horton scaling is a weaker property that addresses the principal branching in a tree; it is a counterpart of the power-law size distribution for systemÕs elements. The stronger Tokunaga self-similarity addresses so-called side-branching; it ensures that different levels of a hierarchy have the same probabilistic structure (in a sense that can be rigorously defined). The solid empirical evidence suggests an existence of a universal self-similarity mechanism and prompts the question: What probability models can generate Horton/Tokunaga self-similar trees with a range of parameters? This talk reviews the existing results and recent findings on self-similarity for tree representation of branching, coalescent processes and time series. We show that the essential models, including white noises, random walks, critical Galton-Watson branching and KingmanÕs coalescent produce trees with Tokunaga and/or Horton self-similarity. Our results explain, at least partially, the omnipresence of Tokunaga and Horton structures and suggest a framework for their statistical analysis. The results are illustrated using geophysical applications.
This is a joint work with Yevgeniy Kovchegov (Oregon State U).

## Fall, 2015

### Title and abstract

Tuesday, October 13, 4:00 pm
STAG 112
Yevgeniy Kovchegov
Oregon State University

"Path Coupling and Aggregate Path Coupling"

Abstract. We describe and characterize an extension to the classical path coupling method referred to as aggregate path coupling. In conjunction with large deviations estimates, we use this aggregate path coupling method to prove rapid mixing of Glauber dynamics for a large class of statistical mechanical models, including models that exhibit discontinuous phase transitions which have traditionally been more difficult to analyze rigorously. Specifically, the parameter regions for rapid mixing for the generalized Curie-Weiss-Potts model and the mean-field Blume-Capel model are derived as an application of the aggregate path coupling method.
Joint work with Peter T. Otto of Willamette University.
Tuesday, October 20, 4:00 pm
STAG 112
University College London, UK

"Modeling Lagrangian trajectories as stochastic processes"

Abstract. Fluid dynamics are often modeled and estimated from the Lagrangian perspective. In this talk I will discuss basic stochastic processes that mimic the behavior of Lagrangian trajectories observed in many environments - for example the motion of satellite-tracked ocean surface drifters. Although these basic processes do not exactly replicate the complex behavior of the ocean surface, fitting their parameters to observed data provides useful summaries of structure. The challenge of Lagrangian data is that it moves both in time and space, and as such I will discuss how to extend our stochastic processes to account for the inherent nonstationarity and heterogeneity of ocean dynamics. Finally, I will discuss extensions to our model to account for anisotropy and demonstrate this with application to a quasi-geostrophic turbulence simulation.
Tuesday, November 3, 4:00 pm
STAG 112
Ed Waymire
Oregon State University

"Analysis of a Particular Random Dynamical System"

Abstract. Non-irreducible random dynamical systems arise naturally in many contexts as iid iterated random maps on a metric space. Nonetheless the problem is to sort out the long time behavior. Some special theory will be described in the context of an example from mathematical biology.
This is based on joint work with Scott Peckham at the University of Colorado, Boulder, and Patrick DeLeenheer.
Tuesday, December 1, 4:00 pm
STAG 112
Victor Peñaranda
Visiting International Scholar
Oregon State University

"Geometrical Features in the Statistical Structure of Rainfall Patterns"

Abstract. Understanding rainfall patterns is and will be a challenge for those working in hydrological sciences. The complexity of rainfall resembles turbulence (not phenomenologically), and some models for describing rainfall have been inspired by turbulence. In this lecture, some ideas about the statistical description of rainfall will be presented, and how some of its geometrical features obtained from the fractal geometry can help us to identify symmetries in the physical process of rainfall. These symmetries are indexed by time and if one identifies the breaking of these symmetries, the space-time dynamics of rainfall could be explained. Some approaches related to the latter statement will also be discussed.

## Winter, 2016

### Title and abstract

Tuesday, January 19, 4:00 pm
WNGR 201
Bartłomiej Siudeja
University of Oregon

"Transition densities and trace estimates for a broad class of Levy processes"

Abstract. Transition density of a stochastic process allows one to quantify the dynamics of the process. Yet, except for Brownian motion and a very few special cases, there is no closed formula for the density, which is usually defined via a characteristic function. The problem is exacerbated for killed processes (confined to bounded domains), where even the Brownian case is not explicit.
We will discuss recent progress on bounding the transition probabilities of a class of killed Levy processes using geometric properties of their domains. We will use these to estimate their traces, the quantities revealing so-called heat invariants. In the classical, Brownian case, the first two invariants are the volume and the surface area of the domain. Surprisingly, we will find the same simple quantities in traces of very general Levy processes.
Tuesday, January 26, 4:00 pm
WNGR 201
Albert M. Fisher
Department of Mathematics
University of São Paulo, Brazil

"The transition in renewal processes to fractal-like returns and ergodic infinite measures"

Abstract. A renewal process counts the number of "events" in the case where the gaps between successive events are independent of each other and are distributed in the same way. One example is the number of burnt-out light bulbs up to a certain time; another is the number of returns to a given state of a countable state Markov process, for example the returns to zero of a random walk. Given the gap distribution, one can calculate the expected return time; there is a dichotomy between this (the first moment) being finite or infinite. One can make a finer distinction, considering the least moment alpha that is finite. Of particular importance is the second moment (the variance); whether or not this is finite subdivides the finite expectation region in two. Making the assumption of regularly varying tails, the result is three "phases" of asymptotic behavior: Gaussian, stable and Mittag-Leffler.
From the point of ergodic theory, the renewal process counts the number of returns to a subset of finite measure, and as we transition through the three phases, we can observe the transition from finite to infinite invariant measure. Precisely, we show that in each case one has asymptotically self-similar returns, stated as an almost-sure invariance principle in log density. For the infinite measure case this is interpreted as fractal-like return structure, leading to an order-two ergodic theorem.
Tuesday, February 9, 4:00 pm
WNGR 201
Li Chen
Intel Corporation, Oregon

"Discovering Insights from Graphs -- When Pattern Recognition Meets Social Network, Neural Connectomes and Digital Marketing"

Abstract. A graph is a representation of a collection of interacting objects. The field of pattern recognition developed significantly in the 1960s, and the field of random graph inference has enjoyed much recent progress in both theory and application. This talk focuses on pattern recognition in the context of a particular family of random graphs, namely the stochastic blockmodels, from the two main perspectives of single graph inference and joint graph inference, as well as its applications in social network, neural connectomes and digital marketing.
Single graph inference is the performance of statistical inference on one single observed graph. Given a single graph realized from a stochastic blockmodel, we here consider the specific exploitation tasks of vertex classification, clustering, and nomination. The theoretical guarantees of these methods are proved and their effectiveness are demonstrated in simulation as well as real datasets including communication network, online advertising, and neural connectomes. We are also concerned with joint graph inference, which involves the joint space of multiple graphs. Specifically, given two graphs, we consider the tasks of seeded graph matching for large graphs and joint vertex classification. The methodologies are shown to discover signals in the joint geometry of diffusion tensor MRI and the Caenorhabditis elegans neural connectomes.
Tuesday, March 1, 4:00 pm
WNGR 201
José Javier Cerda Hernández
Oregon State University
(visiting from Brazil)

"Critical probability of percolation on random causal triangulations"

Abstract. In this work we study bond percolation on random causal triangulations. While in the sub-critical regime there is no phase transition, we show that for percolation on critical random causal triangulations there exists a non-trivial phase transition and we compute an upper bound for the critical probability. Furthermore, the critical value is shown to be almost surely constant.
Tuesday, March 8, 4:00 pm
WNGR 201
Huanqun Jiang
Oregon State University

"Dividend optimization problem in insurance mathematics: a historical review"

Abstract. In this talk, we will introduce a classical optimization problem in insurance, called optimal dividend strategies. The problem was first considered by Bruno De Finetti in 1957, and it had been intensively studied to this day. In the past, the researchers developed several types of dividend strategies which maximize the expectation of total discounted dividends. One of these, called the barrier type fascinates me. The diffusion approximation model for surplus process was proved to have the optimality of barrier strategy under some technical conditions. Later, the optimal barrier strategy was proved possible for the compound Poisson case (Gerber and Shiu 2006). However the remarkable results (Loeffen 2008, Kyprianou 2010) for general negative spectrally Levy process provide the sufficient conditions of optimality for the barrier strategy. Specifically, the scale functions for a Levy process should be convex beyond some point, or Levy measure should be log-convex. This was later extended further to other cases like refracted Levy processes. Recently, there had been a rising interest in the models whose discounting factor (interest rate) evolves as a classical stochastic process. The result of Eisenberg 2015 in this regard will be discussed, and two simple extensions and open questions will be given.

## Spring, 2016

### Title and abstract

Tuesday, April 5, 3:30 pm
GILK 115
University of Washington

"The Erdos-Szusz-Turan distribution"

Abstract. Dirichlet's theorem on diophantine approximation states that for any irrational number a, there are infinitely many rationals p/q so that q|aq-p| < 1. Erdos-Szusz-Turan asked the question of the probability that this estimate could be improved to q|aq-p|< A, with q in a fixed range [N, cN], and the behavior of this probability as N grows. We answer their question and provide a wide-ranging generalization to the setting of equivariant point processes. This is joint work with Anish Ghosh.
Tuesday, May 3, 4:00 pm
GILK 115
Yevgeniy Kovchegov
Oregon State University

"Horton-Strahler ordering and Tokunaga indexing in stochastic processes"

Abstract. We introduce a class of stochastic processes that we call hierarchical branching processes. By construction, the processes satisfy the Tokunaga, and hence Horton, self-similarity constraints. Taking the limit of averaged stochastic dynamics, we obtain the deterministic system of differential equations that describe the temporal dynamics of a Tokunaga branching system. In particular, we study the averaged tree width function to establish a phase transition in the Tokunaga dynamics that separates fading and explosive branching. We then describe a class of critical hierarchical branching processes (that happen at the phase transition boundary) that includes as a special case the celebrated critical Galton-Watson branching process. We illustrate efficiency of the critical hierarchical branching processes in describing diverse observed dendritic structures, and discuss the related critical phenomena from the point of view of respective applications.
Joint work with Ilya Zaliapin (University of Nevada Reno).
Tuesday, May 24, 4:00 pm
GILK 115
Amber Puha
California State University, San Marcos

"Diffusion Limits for Shortest Remaining Processing Time Queues under Nonstandard Spatial Scaling"

Abstract. In a shortest remaining processing time (SRPT) queue, the job that requires the least amount of processing time is preemptively served first. One effect of this is that the queue length is small in comparison to the total amount of work in the system (measured in units of processing time). In the case of processing time distributions with unbounded support, the queue length is so small that the sequence of queue length processes associated with a sequence of SRPT queues, rescaled with standard functional central limit theorem scaling and satisfying standard heavy traffic conditions, converge in distribution to the process that is identically equal to zero. This happens despite the fact that in this same regime the rescaled workload processes converge to a non-degenerate reflected Brownian motion. In particular, the queue length process is of smaller order magnitude than the workload process. In the case of processing time distributions that satisfy a rapid variation condition, we implement an alternative, unconventional spatial scaling that leads to a non-trivial limit for the queue length process. This result quantifies this order of magnitude difference between queue length and workload processes. We illustrate this result for Weibull processing time distributions.

## Fall, 2016

### Title and abstract

Tuesday, October 11, 4:00 pm
WNGR 285
Andrey Sarantsev
Department of Statistics and Applied Probability
University of California, Santa Barbara

"Convergence rates for reflected jump-diffusions on the half-line"

Abstract. An explicit rate of exponential convergence of a continuous-time Markov process to its stationary distribution is hard to find or estimate. However, it is possible for a reflected diffusion process with jumps on the positive half-line. This continues the work of Lund, Meyn, Tweedie (1996).

## Winter, 2017

### Title and abstract

Tuesday, January 10, 4:00 pm
BEXL 321
Patrick Waters
Temple University

"Random matrices and the Stochastic Bessel Operator"

Abstract. The eigenvalues of random matrix "invariant ensembles" can be understood as interacting particle systems, but with temperature restricted to three possible values. Recently "beta ensembles", which extend temperature to all positive values, have been well studied. It has been conjectured that the extremal eigenvalues of a large beta ensemble random matrix with a "hard edge" should be governed by the Stochastic Bessel Operator (SBO). We prove that this conjecture holds when the external field is a polynomial satisfying a convexity condition and β≥1. The law of a smallest SBO eigenvalue gives a two parameter generalization of the famous Tracy-Widom distribution which can be observed in the fluctuations of a spreading coffee stain, the longest increasing subsequence of a random permutation, etc. Joint work with Brian Rider.
Tuesday, February 14, 4:00 pm
BEXL 321
William Felder
Oregon State University

"Distinguished path analysis for continuous-time branching processes: a framework and applications"

Abstract. In this talk we will consider a rubric, laid out by Hardy and Harris, under which many earlier formulations of distinguished path analysis (or "spine techniques") for branching processes are unified.
It has been known for some time that there is a connection between single-particle martingales and certain additive martingales for the corresponding branching processes. This connection is made explicit under the Hardy/Harris framework, where each is seen to be the projection of a single, more general martingale onto different sub sigma-algebras. We will also see a nice, intuitive formulation of the martingale change of measure that results in the typical alterations along the spine: namely a change in the drift, a change in the offspring distribution ("size biasing"), and a change in the reproductive rate.
The setting and machinery laid out by Hardy and Harris are quite elegant, and create a very natural vantage from which to approach distinguished path analysis. The power of their framework will be demonstrated through consideration of example applications.

## Spring, 2017

### Title and abstract

Tuesday, April 18, 4:00 pm
GILK 100
Yevgeniy Kovchegov
Oregon State University

"Coalescence and minimal spanning trees of irregular graphs"

Abstract. We devise a method of finding the limiting mean length of a minimal spanning tree for a random graph via the Smoluchowski coagulation equations for the corresponding coalescent process. In particular, we use this approach for finding the limiting mean length of a minimal spanning tree for the Erdos-Renyi random graph on an asymmetric bipartite graph, producing a completely new formula yet consistent with the previously known formula for the symmetric bipartite graph.

Joint work with Peter T. Otto of Willamette University and Anatoly Yambartsev of University of São Paulo.
Tuesday, May 9, 4:00 pm
GILK 100
Sharmodeep Bhattacharyya
Statistics Department
Oregon State University

"Spectral Clustering for Dynamic Block Models"

Abstract. One of the most common and crucial aspects of many network data sets is the dependence of network link structure on time. In this work, we consider the problem of finding a common clustering structure in time-varying networks. We also propose an extension of the static version of nonparametric latent variable models into the dynamic setting and use special cases of the dynamic models to justify the spectral clustering methods. We consider two extensions of spectral clustering methods to dynamic settings and give theoretical guarantee that the spectral clustering methods produce consistent community detection in case of both dynamic stochastic block model and dynamic degree-corrected block model. The methods are shown to work under sufficiently mild conditions on the number of time snapshots of networks and also if the networks at each time snapshot are sparse and networks at most time snapshots are below community detectability threshold. We show the validity of the theoretical results via simulations too. (Joint work with Shirshendu Chatterjee, CUNY)
Tuesday, May 30, 4:00 pm
GILK 100
Mathew Titus
Oregon State University

"Mixing times for diffusive systems"

Abstract. Mixing times measure how quickly a Markov process approaches its stationary distribution. Often computer scientists and statistical physicists use Markov chains to model networks with n nodes or physical systems with n particles and are interested in the mixing time as a function of large n. In this talk we explore a novel and general method of computing the mixing time asymptotics for one-dimensional diffusive Markov chains. The related question of which chains exhibit cutoff phenomena and with what window-size is also discussed.

## Fall, 2017

### Title and abstract

Tuesday, October 10, 4:00 pm
BEXL 207
Yevgeniy Kovchegov
Oregon State University

"Random self-similar trees: dynamical pruning and its applications to inviscid Burgers equations"

Abstract. We introduce generalized dynamical pruning on rooted binary trees with edge lengths. The pruning removes parts of a tree T, starting from the leaves, according to a pruning function defined on subtrees within T. The generalized pruning encompasses a number of previously studied discrete and continuous pruning operations, including the tree erasure and Horton pruning. For example, a finite critical binary Galton-Watson tree with exponential edge lengths is invariant with respect to the generalized dynamical pruning for arbitrary admissible pruning function. We will discuss an application in which we examine a one dimensional inviscid Burgers equation with a piece-wise linear initial potential with unit slopes. The Burgers dynamics in this case is equivalent to a generalized pruning of the level set tree of the potential, with the pruning function equal to the total tree length. We give a complete description of the Burgers dynamics for the Harris path of a critical binary Galton-Watson tree with i.i.d. exponential edge lengths.

This work was done in collaboration with Ilya Zaliapin (University of Nevada Reno) and Maxim Arnold (University of Texas at Dallas).
Tuesday, October 31, 4:00 pm
BEXL 207
James Watson
CEOAS, Oregon State University

"Anticipating and Managing Risk in Marine Social-ecological Systems"

Abstract. In this presentation I introduce a new approach to managing risk in marine food-producing systems -- fisheries and aquaculture -- based on weather index and cooperative insurance. The mathematical framework is explained and then results are shown from its application to a case-study in Myanmar, where empirical data has been collected. I will also briefly introduce early work from a new project focusing on Manifold Learning, a new approach to studying the multilevel dynamics of complex systems.
Tuesday, November 21, 4:00 pm
BEXL 207
Ed Waymire
Oregon State University

"When 4th Moments Are Enough"

Abstract. This talk concerns a somewhat innocent question motivated by an observation concerning the use of Chebyshev bounds on sample estimates of p in the binomial distribution with parameters n, p. Namely, what moment order produces the best Chebyshev estimate of p? If Sn(p) has a binomial distribution with parameters n, p, then it is readily observed that argmax0 ≤ p ≤ 1ESn2(p) = argmax0 ≤ p ≤ 1np(1-p) = 1/2, and ESn2 = n/4. Rabi Bhattacharya observed (personal communication) that while the second moment Chebyshev sample size for a 95% confidence estimate within ±5 percentage points is n = 2000, the fourth moment yields the substantially reduced polling requirement of n = 775. Why stop at fourth moment? Is the argmax achieved at p = 1/2 for higher order moments and, if so, does it help, and can one easily compute the moment bound ESn2m(1/2)? As captured by the title of this talk, answers to these questions lead to a simple rule of thumb for best choice of moments in terms of an effective sample size for Chebyshev concentration inequalities. This talk is based on joint work with Chris Jennings-Shafer and Dane Skinner.