Asymptotics of unitary and orthogonal matrix integrals

被引：29

作者：

Collins, Benoit ^{[2
,3
]}

Guionnet, Alice ^{[4
]}

Maurel-Segala, Edouard ^{[1
]}

机构：

[1] Univ Paris 06, LPMA, F-75013 Paris, France

[2] Univ Lyon 1, CNRS, ICJ, F-69622 Villeurbanne, France

[3] Univ Ottawa, Dept Math Stat, Ottawa, ON K1N 6N5, Canada

[4] ENS Lyon, UMPA, CNRS, UMR 5669, F-69007 Lyon, France

来源：

ADVANCES IN MATHEMATICS | 2009年 / 222卷 / 01期

基金：

加拿大自然科学与工程研究理事会;

关键词：

Matrix integrals; HCIZ integral; Schwinger-Dyson equation; 2ND-ORDER FREENESS; LARGE DEVIATIONS; FLUCTUATIONS; ENUMERATION;

D O I：

10.1016/j.aim.2009.03.019

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

In this paper. we prove that in small parameter regions, arbitrary unitary matrix integrals converge in the large N limit and match their formal expansion. Secondly we give a combinatorial model for our matrix integral asymptotics and investigate examples related to free probability and the HCIZ integral. Our convergence result also leads us to new results of smoothness of microstates. We finally generalize our approach to integrals over the orthogonal group. (C) 2009 Elsevier Inc. All rights reserved.

引用

页码：172 / 215

页数：44

共 35 条

[1] On the 1/n expansion for some unitary invariant ensembles of random matrices [J].

Albeverio, S ;

Pastur, L ;

Shcherbina, M .

COMMUNICATIONS IN MATHEMATICAL PHYSICS, 2001, 224 (01) :271-305

[2] MATRIX MODEL-CALCULATIONS BEYOND THE SPHERICAL LIMIT [J].

AMBJORN, J ;

CHEKHOV, L ;

KRISTJANSEN, CF ;

MAKEENKO, Y .

NUCLEAR PHYSICS B, 1993, 404 (1-2) :127-172

[3]

[Anonymous], 2008, NEW YORK J MATH

[4]

[Anonymous], 1998, Ecole d'Ete de Probabilites de Saint-Flour XXVIII-1998

[5]

[Anonymous], MATH SURVEYS MONOGR

[6]

[Anonymous], 2006, LECT COMBINATORICS F, DOI DOI 10.1017/CBO9780511735127

[7]

[Anonymous], 1992, American Mathematical Soc.

[8]

BenArous G, 1997, PROBAB THEORY REL, V108, P517

[9]

BERGERE M, 2008, ARXIV08054482

[10] Logarithmic Sobolev inequalities, matrix models and free entropy [J].

Biane, P .

ACTA MATHEMATICA SINICA-ENGLISH SERIES, 2003, 19 (03) :497-506

← 1 2 3 4 →