### 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) |
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Pages (from-to) | 31-81 |

Number of pages | 51 |

Journal | Communications in Mathematical Physics |

Volume | 278 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2008 |

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### 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.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 278, no. 1, pp. 31-81. https://doi.org/10.1007/s00220-007-0395-z

}

TY - JOUR

T1 - Random matrices, graphical enumeration and the continuum limit of toda lattices

AU - Ercolani, Nicholas M

AU - Mclaughlin, Kenneth D T

AU - Pierce, V. U.

PY - 2008/2

Y1 - 2008/2

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=38049053794&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38049053794&partnerID=8YFLogxK

U2 - 10.1007/s00220-007-0395-z

DO - 10.1007/s00220-007-0395-z

M3 - Article

AN - SCOPUS:38049053794

VL - 278

SP - 31

EP - 81

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -