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 - γ K s ) , 0 < γ < 1 , of the integrable Fredholm operator K s acting on the interval (−1, 1) with kernel $${K_s(\lambda, \mu)= \frac{\sin s(\lambda - \mu)}{\pi (\lambda-\mu)}}$$ K s ( λ , μ ) = sin s ( 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 = - 1 2 ln ( 1 - γ ) supported on an interval of length $${\frac{2s}{\pi}}$$ 2 s π . We evaluate, in particular, the double scaling limit of $${\det(I-\gamma K_s)}$$ det ( I - γ K s ) 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 = v s = - 1 2 s ln ( 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).

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