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Almost sure weak convergence for the generalized orthogonal ensemble.

Research output: Contribution to journalJournal article

Published

<mark>Journal publication date</mark>1/10/2001
<mark>Journal</mark>Journal of Statistical Physics
Issue1-2
Volume105
Number of pages27
Pages309-335
<mark>Original language</mark>English

Abstract

The generalized orthogonal ensemble satisfies isoperimetric inequalities analogous to the Gaussian isoperimetric inequality, and an analogue of Wigner's law. Let v be a continuous and even real function such that V(X)=tracev(X)/n defines a uniformly p-convex function on the real symmetric n×n matrices X for some p2. Then (dX)=e –V(X) dX/Z satisfies deviation and transportation inequalities analogous to those satisfied by Gaussian measure(6, 27), but for the Schatten c p norm. The map, that associates to each XM s n () its ordered eigenvalue sequence, induces from a measure which satisfies similar inequalities. It follows from such concentration inequalities that the empirical distribution of eigenvalues converges weakly almost surely to some non-random compactly supported probability distribution as n.

Bibliographic note

RAE_import_type : Journal article RAE_uoa_type : Pure Mathematics