Abstract
For beta-ensembles with convex polynomial potentials, we prove a large deviation principle for the empirical spectral distribution seen from the rightmost particle. This modified spectral distribution was introduced by Perret and Schehr (J. Stat. Phys. 2014) to study the crowding near the maximal eigenvalue, in the case of the GUE. We prove also convergence of fluctuations.
Highlights
In random matrix models, the most popular statistics is the empirical spectral distribution (ESD)
The second one is that the large deviation Principle (LDP) is weak i.e. we do not have a large deviation upperbound for closed sets but only for compact sets
This implies that we could not deduce the convergence to the limit from the LDP as usual
Summary
The most popular statistics is the empirical spectral distribution (ESD). Where λ(1) < λ(2) < · · · < λ(N) are the eigenvalues of MN ranked increasingly They considered the Gaussian case with the Dyson values β = 1, 2, 4 and made a complete study of EμN , in the limit N → ∞ both in the bulk and at the edge. We prove that the family of distributions of (μN )N satisfies the LDP with speed N 2 and a “new" rate function which we call IVDOS, referring to the name “Density of States near the maximum" given by Perret and Schehr to EμN. The second one is that the LDP is weak i.e. we do not have a large deviation upperbound for closed sets but only for compact sets This implies that we could not deduce the convergence to the limit from the LDP as usual.
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