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Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle. (English) Zbl 1077.60079

One problem in the random matrix theory is the estimation of the probability for a given interval to be free from the eigenvalues. In the Gaussian unitary ensemble, this probability for any interval of length \(2s\) in the bulk scaling limit is equal to the following Fredholm determinant: \(D(s)=\text{ det} [I-K],\) where \(K\) is the integral \[ (Kg)(x)=\int^{2s}_0\frac{\sin(x-y)}{\pi(x-y)}g(y)dy. \] It is proven that \[ \ln D(s)=-\frac{s^2}{2}-\frac{1}{2}\ln s+\frac{1}{12}\ln 2+3\zeta'(-1)+O(1/s), \] for \(s\to\infty\) where \(\zeta'(x)\) is the derivative of Riemann’s zeta function.

MSC:

60K40 Other physical applications of random processes
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
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