Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Almost sure weak convergence for the generalized orthogonal ensemble.
AU - Blower, Gordon
N1 - RAE_import_type : Journal article RAE_uoa_type : Pure Mathematics
PY - 2001/10/1
Y1 - 2001/10/1
N2 - 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.
AB - 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.
KW - random matrices - transportation - isoperimetric inequality - statistical mechanics
U2 - 10.1023/A:1012294429641
DO - 10.1023/A:1012294429641
M3 - Journal article
VL - 105
SP - 309
EP - 335
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
SN - 0022-4715
IS - 1-2
ER -