### 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 language | English (US) |
---|---|

Pages (from-to) | 31-81 |

Number of pages | 51 |

Journal | Communications in Mathematical Physics |

Volume | 278 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2008 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics