Asymptotic forms for hard and soft edge general $\beta$ conditional gap probabilities

TL;DR

This paper extends Dyson's log-gas formalism to compute asymptotic gap probabilities at the edges of random matrix beta-ensembles, introducing a potential drop term for accuracy with multiple eigenvalues.

Contribution

It modifies the existing formalism by replacing the entropy term with a potential drop term, enabling accurate asymptotic expansions for conditioned gap probabilities with multiple eigenvalues.

Findings

Derived conjectured asymptotic expansions for gap probabilities including multiple eigenvalues.

Modified the formalism to incorporate a potential drop term for consistency with known results.

Established an asymptotic duality relating beta to 4/beta.

Abstract

An infinite log-gas formalism, due to Dyson, and independently Fogler and Shklovskii, is applied to the computation of conditioned gap probabilities at the hard and soft edges of random matrix $\beta$-ensembles. The conditioning is that there are $n$ eigenvalues in the gap, with $n \ll |t|$, $t$ denoting the end point of the gap. It is found that the entropy term in the formalism must be replaced by a term involving the potential drop to obtain results consistent with known asymptotic expansions in the case $n=0$. With this modification made for general $n$, the derived expansions - which are for the logarithm of the gap probabilities - are conjectured to be correct up to and including terms O$(\log|t|)$. They are shown to satisfy various consistency conditions, including an asymptotic duality formula relating $\beta$ to $4/\beta$.