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 0\), or if the variance of the trace is O(1/n2), 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)