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

Ionel Popescu

• We give a short proof of an extension of the free Talagrand transportation cost inequality to the semicircular which was originally proved in \cite{BV}. The proof is based on a convexity argument and is in the spirit of the original Talagrand's approach for the classical counterpart from \cite{T}. We also discuss the convergence, fluctuations and large deviations of the energy of the eigenvalues of $\beta$ ensembles, which, as an application of Talagrand inequality gives in particular yet another proof of the convergence of the eigenvalue distribution to the semicircle law.