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.
[Patrick is a former OSU grad student. He completed his MS degree in Mathematics in 2010 and his PhD at the University of Arizona in 2015. He is currently a postdoc at Temple University.]