Painlevé II in Random Matrix Theory and Related Fields

Two Lax systems for the Painlevé II Equation, and Two Related Kernels in Random Matrix Theory

We consider two Lax systems for the homogeneous Painlevé II equation: one of size 2×2 studied by Flaschka and Newell in the early 1980s, and one of size 4×4 introduced by Delvaux, Kuijlaars, and Zhang and Duits and Geudens in the early 2010s. We prove that solutions to the 4×4 system can be derived from those to the 2 × 2 system via an integral transform, and consequently relate the Stokes mult...

Discrete Painlevé Equations and Random Matrix Averages

The τ-function theory of Painlevé systems is used to derive recurrences in the rank n of certain random matrix averages over U (n). These recurrences involve auxilary quantities which satisfy discrete Painlevé equations. The random matrix averages include cases which can be interpreted as eigenvalue distributions at the hard edge and in the bulk of matrix ensembles with unitary symmetry. The re...

Random Matrix Theory over Finite Fields

The first part of this paper surveys generating functions methods in the study of random matrices over finite fields, explaining how they arose from theoretical need. Then we describe a probabilistic picture of conjugacy classes of the finite classical groups. Connections are made with symmetric function theory, Markov chains, Rogers-Ramanujan type identities, potential theory, and various meas...

Introductory Workshop Random Matrix Models and Their Applications Airy Kernel and Painlevé Ii

We prove that the distribution function of the largest eigenvalue in the Gaussian Unitary Ensemble (GUE) in the edge scaling limit is expressible in terms of Painlevé II. Our goal is to concentrate on this important example of the connection between random matrix theory and integrable systems, and in so doing to introduce the newcomer to the subject as a whole. We also give sketches of the resu...

Random Fields Theory