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 , of the integrable Fredholm operator K (s) acting on the interval (-1, 1) with kernel . This determinant arises in the analysis of a log-gas of interacting particles in the bulk-scaling limit, at inverse temperature , in the presence of an external potential supported on an interval of length . We evaluate, in particular, the double scaling limit of as and , in the region , for any fixed . This problem was first considered by Dyson (Chen Ning Yang: A Great Physicist of the Twentieth Century. International Press, Cambridge, pp. 131-146, 1995).