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

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