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−γKs),0<γ<1 , of the integrable Fredholm operator K s acting on the interval (−1, 1) with kernel Ks(λ,μ)=sins(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=−12ln(1−γ) supported on an interval of length 2sπ . We evaluate, in particular, the double scaling limit of det(I−γKs) as s→∞ and γ↑1 , in the region 0≤κ=vs=−12sln(1−γ)≤1−δ , for any fixed 0<δ<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).