OSU PROBABILITY SEMINARDepartment of Mathematics |
Day/Time/Room |
Speaker |
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 7, 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 here, 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 overall formulation is quite elegant, and its power will be demonstrated through consideration of example applications. |