Journal article

Finite Size Effects for Spacing Distributions in Random Matrix Theory: Circular Ensembles and Riemann Zeros

Folkmar Bornemann, Peter J Forrester, Anthony Mays

Studies in Applied Mathematics | Wiley | Published : 2017

Abstract

According to Dyson's threefold way, from the viewpoint of global time reversal symmetry, there are three circular ensembles of unitary random matrices relevant to the study of chaotic spectra in quantum mechanics. These are the circular orthogonal, unitary, and symplectic ensembles, denoted COE, CUE, and CSE, respectively. For each of these three ensembles and their thinned versions, whereby each eigenvalue is deleted independently with probability 1 − ξ, we take up the problem of calculating the first two terms in the scaled large N expansion of the spacing distributions. It is well known that the leading term admits a characterization in terms of both Fredholm determinants and Painlevé tra..

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Grants

Awarded by DFG-Collaborative Research Center, "Discretization in Geometry and Dynamics"


Funding Acknowledgements

We are grateful to A. Odlyzko for providing us with the Riemann zero data set. We would also like to thank the referees for their thoughtful comments. The research of F.B. was supported by the DFG-Collaborative Research Center, TRR 109, "Discretization in Geometry and Dynamics." The research of P.J.F. and A.M. is part of the program of study supported by the ARC Centre of Excellence for Mathematical & Statistical Frontiers.