On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential II

TL;DR

This paper provides rigorous proofs for the asymptotic behavior of a determinant related to a sine kernel operator, extending previous work on its properties in the bulk scaling limit with a varying external potential.

Contribution

It offers the first complete proofs of key asymptotic results for the determinant of a sine kernel operator with a parameter, advancing understanding of its behavior in the bulk scaling limit.

Findings

Proved asymptotic formulas for the determinant involving the sine kernel.

Established rigorous bounds and behavior of the operator in the bulk limit.

Extended previous results with detailed proofs and analysis.

Abstract

In this paper we continue our analysis \cite{BDIK} of the determinant $\det(I-\gamma K_s),\gamma\in(0,1)$ where $K_s$ is the trace class operator acting in $L^2(-1,1)$ with kernel $K_s(\lambda,\mu)=\frac{\sin s(\lambda-\mu)}{\pi(\lambda-\mu)}$. In \cite{BDIK} various key asymptotic results were stated and utilized, but without proof: Here we provide the proofs (see Theorem 1.2 and Proposition 1.3 below).