Journal article

# Diffusion processes and the asymptotic bulk gap probability for the real Ginibre ensemble

Peter J Forrester

Journal of Physics A: Mathematical and Theoretical | IOP PUBLISHING LTD | Published : 2015

#### Abstract

It is known that the bulk scaling limit of the real eigenvalues for the real Ginibre ensemble is equal in distribution to the rescaled t → ∞ limit of the annihilation process A + A → Ø. Furthermore, deleting each particle at random in the rescaled t → ∞ limit of the coalescence process A + A → A, a process equal in distribution to the annihilation process results. We use these inter-relationships to deduce from the existing literature the asymptotic small and large distance form of the gap probability for the real Ginibre ensemble. In particular, the leading form of the latter is shown to be equal to exp(-(ζ(3/2)/(2√2π))s), where s denotes the gap size and ζ(z) denotes the Riemann zeta funct..

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#### Grants

#### Funding Acknowledgements

Financial support for this work from the Australian Research Council is acknowledged. I thank C Beenakker for sending me a copy of [4] and correspondence which pointed out the numerical studies therein of the spacing distribution for a certain ensemble of matrices related to the real Ginibre ensemble. I also thank J Edge for going to the trouble of comparing the asymptotic formula of this work against large scale simulation data extending [4], and by so doing identifying an error in the reporting of the value of c<INF>2</INF> in the original version of this paper.