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

TL;DR

This paper analyzes the asymptotic behavior of a determinant related to a log-gas model in the bulk scaling limit, focusing on the double scaling limit as parameters grow large, extending Dyson’s earlier work.

Contribution

It provides a detailed evaluation of the double scaling limit of a Fredholm determinant in a log-gas model with a varying external potential, extending previous theoretical results.

Findings

Asymptotic formula for the determinant in the double scaling limit.

Connection to Dyson's earlier analysis of similar models.

Insights into the behavior of log-gases with external potentials.

Abstract

We study the determinant $\det(I-\gamma K_s), 0<\gamma <1$, of the integrable Fredholm operator $K_s$ acting on the interval $(-1,1)$ with kernel $K_s(\lambda, \mu)= \frac{\sin s(\lambda - \mu)}{\pi (\lambda-\mu)}$. This determinant arises in the analysis of a log-gas of interacting particles in the bulk-scaling limit, at inverse temperature $\beta=2$, in the presence of an external potential $v=-\frac{1}{2}\ln(1-\gamma)$ supported on an interval of length $\frac{2s}{\pi}$. We evaluate, in particular, the double scaling limit of $\det(I-\gamma K_s)$ as $s\rightarrow\infty$ and $\gamma\uparrow 1$, in the region $0\leq\kappa=\frac{v}{s}=-\frac{1}{2s}\ln(1-\gamma)\leq 1-\delta$, for any fixed $0<\delta<1$. This problem was first considered by Dyson in \cite{Dy1}.