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

Abstract

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

49 pages, 15 figures. Version 2 contains an extended introduction and corrects typos