Variance Calculations and the Bessel Kernel

TL;DR

This paper derives exact formulas for the variance of linear statistics in the Laguerre ensemble of matrices, focusing on limits as matrix size and parameter alpha grow large, relevant for quantum transport applications.

Contribution

It provides new exact expressions for the asymptotic variance in the Laguerre ensemble, including the challenging limit as alpha approaches infinity.

Findings

Exact formulas for lim_{N->infinity} var_N f

Exact formulas for lim_{alpha->infinity}lim_{N->infinity} var_N f

Results applicable to quantum transport problems

Abstract

In the Laguerre ensemble of N x N (positive) hermitian matrices, it is of interest both theoretically and for applications to quantum transport problems to compute the variance of a linear statistic, denoted var_N f, as N->infinity. Furthermore, this statistic often contains an additional parameter alpha for which the limit alpha->infinity is most interesting and most difficult to compute numerically. We derive exact expressions for both lim_{N->infinity} var_N f and lim_{alpha->infinity}lim_{N->infinity} var_N f.