Asymptotics of Muttalib-Borodin determinants with Fisher-Hartwig singularities

TL;DR

This paper derives large-size asymptotics for Muttalib-Borodin determinants with Fisher-Hartwig singularities, providing insights into their statistical properties and establishing several limit theorems.

Contribution

It extends asymptotic analysis of Muttalib-Borodin determinants to arbitrary Fisher-Hartwig singularities for all positive b8, including new limit theorems and rigidity bounds.

Findings

Asymptotics for determinants with Fisher-Hartwig singularities

Central limit theorems for logarithm of characteristic polynomial

Global bulk rigidity upper bounds

Abstract

Muttalib-Borodin determinants are generalizations of Hankel determinants and depend on a parameter $\theta>0$. In this paper, we obtain large $n$ asymptotics for $n \times n$ Muttalib-Borodin determinants whose weight possesses an arbitrary number of Fisher-Hartwig singularities. As a corollary, we obtain asymptotics for the expectation and variance of the real and imaginary parts of the logarithm of the characteristic polynomial, several central limit theorems, and some global bulk rigidity upper bounds. Our results are valid for all $\theta > 0$.