Random matrices, graphical enumeration and the continuum limit of toda lattices

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

In this paper we derive analytic characterizations for and explicit evaluations of the coefficients of the matrix integral genus expansion. The expansion itself arises from the large N asymptotic expansion of the logarithm of the partition function of N × N Hermitian random matrices. Its g th coefficient is a generating function for graphical enumeration on Riemann surfaces of genus g. The case that we particularly consider is for an underlying measure that differs from the Gaussian weight by a single monomial term of degree 2ν. Our results are based on a hierarchy of recursively solvable differential equations, derived through a novel continuum limit, whose solutions are the coefficients we want to characterize. These equations are interesting in their own right in that their form is related to partitions of 2g + 1 and joint probability distributions for conditioned random walks.

Original languageEnglish (US)
Pages (from-to)31-81
Number of pages51
JournalCommunications in Mathematical Physics
Volume278
Issue number1
DOIs
StatePublished - Feb 2008

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enumeration
Toda Lattice
Continuum Limit
Random Matrices
Enumeration
continuums
expansion
partitions
Genus
Coefficient
coefficients
Hermitian matrix
Monomial
logarithms
Riemann Surface
random walk
Joint Distribution
Logarithm
Partition Function
hierarchies

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Random matrices, graphical enumeration and the continuum limit of toda lattices. / Ercolani, Nicholas M; Mclaughlin, Kenneth D T; Pierce, V. U.

In: Communications in Mathematical Physics, Vol. 278, No. 1, 02.2008, p. 31-81.

Research output: Contribution to journalArticle

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