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

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 $s\to \mathrm{\infty }$ and $\gamma \uparrow 1$, in the region $0\le \kappa =\frac{v}{s}=-\frac{1}{2s}\ln (1-\gamma )\le 1-\delta$, for any fixed $0<\delta <1$. This problem was first considered by Dyson in \cite{Dy1}.