Universal Structure and Universal PDE for Unitary Ensembles

TL;DR

This paper unifies the description of unitary invariant random matrix ensembles using Tracy-Widom and Adler-Shiota-Van Moerbeke methods, deriving a universal PDE for spectral gap probabilities that links orthogonal functions, Toda lattice, and isomonodromic deformations.

Contribution

It introduces a universal PDE for unitary ensembles, connecting orthogonal functions, Toda lattice, and Schlesinger equations, providing a unified framework for spectral gap probabilities.

Findings

Derived general recurrence relations for unitary ensembles.

Established universal relations between TW variables and Toda lattice τ-functions.

Presented a single PDE governing spectral gap probabilities for various UE.

Abstract

An attempt is made to describe random matrix ensembles with unitary invariance of measure (UE) in a unified way, using a combination of Tracy-Widom (TW) and Adler-Shiota-Van Moerbeke (ASvM) approaches to derivation of partial differential equations (PDE) for spectral gap probabilities. First, general 3-term recurrence relations for UE restricted to subsets of real line, or, in other words, for functions in the resolvent kernel, are obtained. Using them, simple universal relations between all TW dependent variables and one-dimensional Toda lattice $\tau$-functions are found. A universal system of PDE for UE is derived from previous relations, which leads also to a {\it single independent PDE} for spectral gap probability of various UE. Thus, orthogonal function bases and Toda lattice are seen at the core of correspondence of different approaches. Moreover, Toda-AKNS system provides a common structure of PDE for unitary ensembles. Interestingly, this structure can be seen in two very different forms: one arises from orthogonal functions-Toda lattice considerations, while the other comes from Schlesinger equations for isomonodromic deformations and their relation with TW equations. The simple example of Gaussian matrices most neatly exposes this structure.