Displacement Convexity for the Generalized Orthogonal Ensemble

ISSN： 0022-4715

Source： Journal of Statistical Physics, Vol.116, Iss.5-6, 2004-09, pp. : 1359-1387

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Abstract

The generalized orthogonal ensemble of n × n real symmetric matrices X has probability measure ν_n(dX) = Z_n^-1exp{ −ntrace v(X)}dX where dX is the product of Lebesgue measure on the matrix entries and v(X)≥ (2+δ)log|X| with δ>0. The eigenvalue distribution is concentrated on [−A/2, A/2] 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′′(X)≥ −κ where 3A²κ <1. If v′′(X)≥ κ₀ > 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 $n\rightarrow\infty$ . 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⁾