Toby Shepherd

Universality for Random Matrices With an Edge Spectrum Singularity

In their 2001 paper, Forrester-Witte showed, using Okamoto's tau-function theory, that the moment-generating functional of the GUE with an edge spectrum singularity of order |x-√2|^α is related to the σ-form of the Painlevé-II equation with coefficient α, generalising the result that was known for the edge of spectrum distribution for the GUE. We use a method of orthogonal polynomials to generalise this result to a class unitary ensembles with more general weight functions exp(-n tr(V(M)). In this talk I will introduce some background material on these ensembles and demonstrate firstly how we can obtain all of the asymptotic information for such ensembles by simply considering the large-n asymptotics of the relevant orthogonal polynomials, before introducing the Riemann-Hilbert method for evaluating these asymptotics.