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Dyson’s constant in the asymptotics of the Fredholm determinant of the sine kernel. (English) Zbl 1113.82030

Summary: We prove that the asymptotics of the Fredholm determinant of \(I-K_\alpha\), where \(K_\alpha\) is the integral operator with the sine kernel \(\frac{\sin(x-y)}{\pi(x-y)}\) on the interval \([0,\alpha]\), are given by \[ \log\det(I-K_{2\alpha}) = -\frac{\alpha^2}{2} -\frac{\log\alpha}{4} +\frac{\log 2}{12} + 3\zeta'(-1) + o(1), \qquad \alpha \to \infty. \] This formula was conjectured by Dyson. The proof for the first and second order asymptotics was given by Widom, and higher order asymptotics have also been determined. In this paper we identify the constant (or third order) term, which has been an outstanding problem for a long time.

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82B05 Classical equilibrium statistical mechanics (general)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
15B52 Random matrices (algebraic aspects)
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