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 \({s\rightarrow\infty}\) 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 (Chen Ning Yang: A Great Physicist of the Twentieth Century. International Press, Cambridge, pp. 131–146, 1995).