Let
Ka±K_\alpha^\pm stand for the integral operators with the sine kernels
\fracsin(x-y)p(x-y) ±\fracsin(x+y)p(x+y)\frac{\sin(x-y)}{\pi(x-y)} \pm \frac{\sin(x+y)}{\pi(x+y)} acting on
L
2[0,α]. Dyson conjectured that the asymptotics of the Fredholm determinants of
I-Ka±I-K_\alpha^\pm are given by
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logdet(I-Ka±) = -\fraca24-±\fraca2-\fracloga8+\fraclog224±\fraclog24 +\frac32 z¢(-1)+o(1), \log\det(I-K_{\alpha}^\pm) = -\frac{\alpha^2}{4}\mp \frac{\alpha}{2}-\frac{\log\alpha}{8}+\frac{\log 2}{24}\pm \frac{\log 2}{4} +\frac{3}{2} \zeta'(-1)+o(1),
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as α→∞. In this paper we are going to give a proof of these two asymptotic formulas.
Communicated by J.L. Lebowitz.