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The Widom–Dyson constant for the gap probability in random matrix theory. (English) Zbl 1116.15019

An asymptotic question in the theory of the Gaussian unitary ensemble of random matrices is considered. In the bulk scaling limit, the probability that there are no eigenvalues in the interval \((0,2s)\) is given by \(P_s~=~\text{ det}(I-K_s)\), where \(K_s\) is the trace-class operator with kernel \(K_s(x,y)= \frac{\text{ sin}(x-y)}{\pi (x-y)}\) acting on \(L^2(0, 2s)\).
The asymptotic behaviour of \(P_s\) as \(s \rightarrow \infty\) was determined earlier and of particular interest in an asymptotic expansion is the Widom-Dyson constant \(c_0 = \frac{\text{ 1}}{\text{ 12}}~{\ln 2} + \text{ 3}\zeta' (- \text{ 1})\), where \(\zeta(z)\) is the Riemann zeta-function. A new derivation of this constant which does not rely on certain a priory information is presented. This approach has the potential advantage of being applicable to other problems involving the computation of critical constants, where a priori information may not be available.

MSC:

15B52 Random matrices (algebraic aspects)
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