Poisson Statistics for Matrix Ensembles at Large Temperature

• Published in: Journal of statistical physics Vol. 161; no. 3; pp. 633 - 656
• Main Authors: Benaych-Georges, Florent, Péché, Sandrine
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.11.2015; Springer; Springer Verlag
• Subjects: Mathematical and Computational Physics; Mathematics; Physical Chemistry; Physics; Physics and Astronomy; Probability; Quantum Physics; Statistical Physics and Dynamical Systems; Theoretical; 15A52; Random matrices; Poisson point process; Ensembles; 60F05
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-015-1340-8

In this article, we consider β -ensembles, i.e. collections of particles with random positions on the real line having joint distribution 1 Z N ( β ) | Δ ( λ ) | β e - N β 4 ∑ i = 1 N λ i 2 d λ , in the regime where β → 0 as N → ∞ . We briefly describe the global regime and then consider the local regime. In the case where N β 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 β → ∞ , we prove a partial result in this direction.