Journal of Statistical Physics

, Volume 161, Issue 3, pp 633–656 | Cite as

Poisson Statistics for Matrix Ensembles at Large Temperature

  • Florent Benaych-Georges
  • Sandrine Péché


In this article, we consider \(\beta \)-ensembles, i.e. collections of particles with random positions on the real line having joint distribution
$$\begin{aligned} \frac{1}{Z_N(\beta )}\big |\Delta (\lambda )\big |^\beta e^{- \frac{N\beta }{4}\sum _{i=1}^N\lambda _i^2}\mathrm {d}\lambda , \end{aligned}$$
in the regime where \(\beta \rightarrow 0\) as \(N\rightarrow \infty \). 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 \(N\beta \rightarrow \infty \), we prove a partial result in this direction.


Random matrices \(\beta \)-Ensembles Poisson point process 

Mathematics Subject Classification

15A52 60F05 



The authors thank Alice Guionnet for her contribution to the proof and her useful suggestions for simplifying the arguments. We also thank Paul Bourgade for useful discussions.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.MAP 5, UMR CNRS 8145 - Université Paris DescartesParis Cedex 6France
  2. 2.LPMA, Université Paris DiderotParisFrance

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