Talagrand Inequality for the Semicircular Law and Energy of the Eigenvalues of Beta Ensembles

Details Talagrand Inequality for the Semicircular Law and Energy of the Eigenvalues of Beta Ensembles Talagrand Inequality for the Semicircular Law and Energy of the Eigenvalues of Beta Ensembles

2006-10-27

arxiv.org/abs/math/0610826v3

Mathematics

Classical Analysis and ODEs

This is the final version. There were some typos and minor errors fixed

Scientific paper

We give a short proof of the free analogue of the Talagrand inequality for the transportation cost to the semicircular which was originally proved by Biane and Voiculescu. The proof is based on a convexity argument and is in the spirit of the original Talagrand's proof. We also discuss the convergence, fluctuations and large deviations of the energy of the eigenvalues of beta ensembles, which gives also yet another proof of the convergence of the eigenvalue distribution to the semicircle law.

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