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Diffusion processes and the asymptotic bulk gap probability for the real Ginibre ensemble. (English) Zbl 1335.82011
The main approach of this paper is to deduce the small and large expansions gap probabilities relying on the equivalence in distribution of the bulk scaling limit of the real eigenvalues for the real Ginibre ensemble and the rescaled \(t\to\infty\) limit of the annihilation process \(A+A\to\emptyset\). In particular, it is established that the leading form of the gap probabilities is equal to \(\text{exp}(-(\zeta(3/2)/(2\sqrt{2\pi}))s)\), where \(s\) is the gap size and \(\zeta(z)\) denotes the Riemann zeta function. It is shown how this can be rigorously established using an asymptotic formula for matrix Fredholm operators. A determinant formula is derived for the gap probability in the finite-\(N\) case, and this is used to illustrate the asymptotic formulas against numerical computations.

MSC:
82B30 Statistical thermodynamics
60B20 Random matrices (probabilistic aspects)
15B52 Random matrices (algebraic aspects)
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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