On the analysis of incomplete spectra in random matrix theory through an extension of the Jimbo-Miwa-Ueno differential

TL;DR

This paper extends the Jimbo-Miwa-Ueno differential form to analyze incomplete spectra in random matrix theory, enabling precise asymptotic calculations of distribution functions in complex Wishart ensembles with random eigenvalue removal.

Contribution

It introduces a novel extension of the Jimbo-Miwa-Ueno differential form that facilitates the asymptotic analysis of incomplete spectra in random matrices.

Findings

Derived tail asymptotics for distribution functions in Wishart ensembles

Provided exact constants for asymptotic formulas

Enabled analysis of spectra with randomly removed eigenvalues

Abstract

Several distribution functions in the classical unitarily invariant matrix ensembles are prime examples of isomonodromic tau functions as introduced by Jimbo, Miwa and Ueno (JMU) in the early 1980s \cite{JMU}. Recent advances in the theory of tau functions \cite{ILP}, based on earlier works of B. Malgrange and M. Bertola, have allowed to extend the original Jimbo-Miwa-Ueno differential form to a 1-form closed on the full space of extended monodromy data of the underlying Lax pairs. This in turn has yielded a novel approach for the asymptotic evaluation of isomonodromic tau functions, including the exact computation of all relevant constant factors. We use this method to efficiently compute the tail asymptotics of soft-edge, hard-edge and bulk scaled distribution and gap functions in the complex Wishart ensemble, provided each eigenvalue particle has been removed independently with probability $1-\gamma\in(0,1]$.