Displacement Convexity for the Generalized Orthogonal Ensemble

Abstract

The generalized orthogonal ensemble of n × n real symmetric matrices X has probability measure $\nu _n \left( {dX} \right) = Z_n^{ - 1} \exp \left\{ { - ntracev\left( X \right)} \right\}$ where dX is the product of Lebesgue measure on the matrix entries and $v\left( x \right) \geqslant \left( {2 + \delta } \right)\log \left| x \right|$ with δ>0. The eigenvalue distribution is concentrated on $\left[ {{{ - A} \mathord{\left/ {\vphantom {{ - A} 2}} \right. \kern-\nulldelimiterspace} 2},{A \mathord{\left/ {\vphantom {A 2}} \right. \kern-\nulldelimiterspace} 2}} \right]$ for some A<∞. This paper establishes concentration and transportation inequalities for the distribution of eigenvalues of X under ν _n when v is twice differentiable with $v''\left( x \right) \geqslant - \kappa$ where$3A^2 \kappa < 1$. If$v''\left( x \right) \geqslant \kappa _0 > 0$, or if the variance of the trace is O(1/n ²), then the empirical distribution of eigenvalues converges weakly almost surely to some non-random probability measure on [−A/2, A/2] as $O\left( {\sqrt {\log {N \mathord{\left/ {\vphantom {N {N^2 }}} \right. \kern-\nulldelimiterspace} {N^2 }}} } \right)$. These conditions are satisfied for certain polynomial potentials. The logarithmic energy is displacement convex as a functional on charge distributions, with fixed mean, along the real line. When the trace distribution satisfies a logarithmic Sobolev inequality, or equivalently a quadratic transportation inequality, the joint eigenvalue distributions and the limiting equilibrium measure likewise satisfy quadratic transportation inequalities in the sense of Talagrand.⁽²⁴⁾