Multicritical Schur measures and higher-order analogues of the Tracy-Widom distribution

TL;DR

This paper introduces multicritical Schur measures, revealing new universality classes with higher-order Tracy-Widom distributions characterized by critical exponents 1/(2m+1), extending the classic Tracy-Widom GUE distribution.

Contribution

It defines multicritical Schur measures, connects them to higher-order Airy kernels, and maps them to multicritical unitary matrix models, expanding understanding of edge fluctuations in random partitions.

Findings

New universality classes with critical exponents 1/(2m+1)

Asymptotic distributions via higher order Airy kernels

Exact mapping to multicritical unitary matrix models

Abstract

We introduce multicritical Schur measures, which are probability laws on integer partitions which give rise to non-generic fluctuations at their edge. They are in the same universality classes as one-dimensional momentum-space models of free fermions in flat confining potentials, studied by Le Doussal, Majumdar and Schehr. These universality classes involve critical exponents of the form 1/(2m+1), with m a positive integer, and asymptotic distributions given by Fredholm determinants constructed from higher order Airy kernels, extending the generic Tracy-Widom GUE distribution recovered for m=1. We also compute limit shapes for the multicritical Schur measures, discuss the finite temperature setting, and exhibit an exact mapping to the multicritical unitary matrix models previously encountered by Periwal and Shevitz.