Character expansion method for the first order asymptotics of a matrix integral

• 成果类型: Article
• 署名作者: Guionnet, A; Maïda, M
• 署名单位: Ecole Normale Superieure de Lyon (ENS de LYON); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-004-0403-6
• 发表日期: 2005
• 页码: 539-578
• 关键词: large-n limit; large deviations; variables; entropy

摘要

The estimation of various matrix integrals as the size of the matrices goes to infinity is motivated by theoretical physics, geometry and free probability questions. On a rigorous ground, only integrals of one matrix or of several matrices with simple quadratic interaction (called AB interaction) could be evaluated so far (see e.g. [19], [17] or [9]). In this article, we follow an idea widely developed in the physics literature, which is based on character expansion, to study more complex interaction. In this context, we derive a large deviation principle for the empirical measure of Young tableaux. We then use it to study a matrix model defined in the spirit of the 'dually weighted graph model' introduced in [13], but with a cutoff function such that the matrix integral and its character expansion converge. We prove that the free energy of this model converges as the size of the matrices goes to infinity and study the critical points of the limit.