Large gaps between random eigenvalues

Details Large gaps between random eigenvalues Large gaps between random eigenvalues

2008-10-31

arxiv.org/abs/0811.0007v3

Annals of Probability 2010, Vol. 38, No. 3, 1263-1279

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AOP508 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/09-AOP508

We show that in the point process limit of the bulk eigenvalues of $\beta$-ensembles of random matrices, the probability of having no eigenvalue in a fixed interval of size $\lambda$ is given by \[\bigl(\ kappa_{\beta}+o(1)\bigr)\lambda^{\gamma_{\beta}}\exp\biggl(-{\bet a}{64}\lambda^2+\biggl({\beta}{8}-{1}{4}\biggr)\lambda\biggr)\] as $\lambda\to\infty$, where \[\gamma_{\beta}={1}{4}\biggl({\beta}{2}+{2}{\beta}-3\biggr)\] and $\kappa_{\beta}$ is an undetermined positive constant. This is a slightly corrected version of a prediction by Dyson [J. Math. Phys. 3 (1962) 157--165]. Our proof uses the new Brownian carousel representation of the limit process, as well as the Cameron--Martin--Girsanov transformation in stochastic calculus.

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