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 $${\det(I-\gamma K_s), 0 < \gamma < 1}$$det(I-γKs),0<γ<1 , of the integrable Fredholm operator K_s acting on the interval (−1, 1) with kernel $${K_s(\lambda, \mu)= \frac{\sin s(\lambda - \mu)}{\pi (\lambda-\mu)}}$$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 $${\beta=2}$$β=2 , in the presence of an external potential $${v=-\frac{1}{2}\ln(1-\gamma)}$$v=-12ln(1-γ) supported on an interval of length $${\frac{2s}{\pi}}$$2sπ. We evaluate, in particular, the double scaling limit of $${\det(I-\gamma K_s)}$$det(I-γKs) as $${s\rightarrow\infty}$$s→∞ and $${\gamma\uparrow 1}$$γ↑1 , in the region $${0\leq\kappa=\frac{v}{s}=-\frac{1}{2s}\ln(1-\gamma)\leq 1-\delta}$$0≤κ=vs=-12sln(1-γ)≤1-δ , for any fixed $${0 < \delta < 1}$$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).