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Poisson statistics for matrix ensembles at large temperature

Abstract : In this article, we consider $\beta$-ensembles, i.e. collections of particles with random positions on the real line having joint distribution $$\frac{1}{Z_N(\beta)}|\Delta(\lambda)|^\beta e^{- \frac{N\beta}{4}\sum_{i=1}^N\lambda_i^2}d \lambda,$$ in the regime where $\beta\to 0$ as $N\to\infty$. We briefly describe the global regime and then consider the local regime. In the case where $N\beta$ stays bounded, we prove that the local eigenvalue statistics, in the vicinity of any real number, are asymptotically to those of a Poisson point process. In the case where $N\beta\to\infty$, we prove a partial result in this direction.
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Contributor : Florent Benaych-Georges <>
Submitted on : Friday, June 12, 2015 - 7:06:58 AM
Last modification on : Friday, April 10, 2020 - 5:05:08 PM

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Florent Benaych-Georges, Sandrine Péché. Poisson statistics for matrix ensembles at large temperature. Journal of Statistical Physics, Springer Verlag, 2015, 161 (3), pp.633-656. ⟨10.1007/s10955-015-1340-8⟩. ⟨hal-01163065⟩



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