Moments at the hard edge and Rayleigh functions

TL;DR

This paper investigates inverse power trace moments of the Laguerre ensemble at the hard edge, deriving explicit formulas for classical beta values and exploring limits involving the Bessel zeta function.

Contribution

It provides explicit inverse moment formulas for classical beta values and extends the analysis to general beta using partition sums, including a low-temperature limit involving the Bessel zeta function.

Findings

Explicit inverse moments for beta=1,2,4

Mellin transform formulas for these moments

Moments expressed via Bessel zeta function as beta approaches infinity

Abstract

Motivated by the analogy between spectral moments of random matrices and associated zeta functions, we study inverse power trace moments of the Laguerre ensemble of dimension $N$ and inverse temperature parameter $\beta>0$. We consider a large $N$ regime determined by the low-lying eigenvalues of the ensemble known as the hard edge. In the classical cases $\beta \in \{1,2,4\}$, we obtain explicit results for the inverse moments and extend these to formulae for the corresponding Mellin transforms. In the case of general $\beta>0$, by a result of Fyodorov and Le Doussal, we obtain a different formula for the moments given as a sum over partitions. We use this to consider a low temperature limit where $\beta \to \infty$ as $N \to \infty$. In this limit, we show that the moments are given in terms of the Bessel zeta function.