### Abstract

This paper is concerned with the asymptotic behavior of the free energy for a class of Hermitian random matrix models, with odd degree polynomial potential, in the large N limit. It continues an investigation initiated and developed in a sequence of prior works whose ultimate aim is to reveal and understand, in a rigorous way, the deep connections between correlation functions for eigenvalues of these random matrix ensembles on the one hand and the enumerative interpretations of their matrix moments in terms of map combinatorics (a branch of graph theory) on the other. In doing this we make essential use of the link between the asymptotics of the random matrix partition function and orthogonal polynomials with exponential weight equal to the random matrix potential. Along the way we develop and analyze the continuum limits of both the Toda lattice equations and the difference string equations associated to these orthogonal polynomials. The former are found to have the structure of a hierarchy of near-conservation laws; the latter are a novel semi-classical extension of the traditional string equations. One has these equations for each class of regular maps of a given valence. Our methods apply to regular maps of both even and odd valence, however we focus on the latter since that is the relevant case for this paper. These methods enable us to rigorously determine closed form expressions for the generating functions that enumerate trivalent maps, in general implicitly, but also explicitly in a number of cases.

Original language | English (US) |
---|---|

Pages (from-to) | 267-305 |

Number of pages | 39 |

Journal | Communications in Mathematical Sciences |

Volume | 10 |

Issue number | 1 |

State | Published - Mar 2012 |

### Fingerprint

### Keywords

- Conservation law hierarchies
- Map enumeration
- Motzkin paths
- Random matrices
- String equations
- Toda lattice

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications in Mathematical Sciences*,

*10*(1), 267-305.

**The continuum limit of Toda lattices for random matrices with odd weights.** / Ercolani, Nicholas M; Pierce, Virgil U.

Research output: Contribution to journal › Article

*Communications in Mathematical Sciences*, vol. 10, no. 1, pp. 267-305.

}

TY - JOUR

T1 - The continuum limit of Toda lattices for random matrices with odd weights

AU - Ercolani, Nicholas M

AU - Pierce, Virgil U.

PY - 2012/3

Y1 - 2012/3

N2 - This paper is concerned with the asymptotic behavior of the free energy for a class of Hermitian random matrix models, with odd degree polynomial potential, in the large N limit. It continues an investigation initiated and developed in a sequence of prior works whose ultimate aim is to reveal and understand, in a rigorous way, the deep connections between correlation functions for eigenvalues of these random matrix ensembles on the one hand and the enumerative interpretations of their matrix moments in terms of map combinatorics (a branch of graph theory) on the other. In doing this we make essential use of the link between the asymptotics of the random matrix partition function and orthogonal polynomials with exponential weight equal to the random matrix potential. Along the way we develop and analyze the continuum limits of both the Toda lattice equations and the difference string equations associated to these orthogonal polynomials. The former are found to have the structure of a hierarchy of near-conservation laws; the latter are a novel semi-classical extension of the traditional string equations. One has these equations for each class of regular maps of a given valence. Our methods apply to regular maps of both even and odd valence, however we focus on the latter since that is the relevant case for this paper. These methods enable us to rigorously determine closed form expressions for the generating functions that enumerate trivalent maps, in general implicitly, but also explicitly in a number of cases.

AB - This paper is concerned with the asymptotic behavior of the free energy for a class of Hermitian random matrix models, with odd degree polynomial potential, in the large N limit. It continues an investigation initiated and developed in a sequence of prior works whose ultimate aim is to reveal and understand, in a rigorous way, the deep connections between correlation functions for eigenvalues of these random matrix ensembles on the one hand and the enumerative interpretations of their matrix moments in terms of map combinatorics (a branch of graph theory) on the other. In doing this we make essential use of the link between the asymptotics of the random matrix partition function and orthogonal polynomials with exponential weight equal to the random matrix potential. Along the way we develop and analyze the continuum limits of both the Toda lattice equations and the difference string equations associated to these orthogonal polynomials. The former are found to have the structure of a hierarchy of near-conservation laws; the latter are a novel semi-classical extension of the traditional string equations. One has these equations for each class of regular maps of a given valence. Our methods apply to regular maps of both even and odd valence, however we focus on the latter since that is the relevant case for this paper. These methods enable us to rigorously determine closed form expressions for the generating functions that enumerate trivalent maps, in general implicitly, but also explicitly in a number of cases.

KW - Conservation law hierarchies

KW - Map enumeration

KW - Motzkin paths

KW - Random matrices

KW - String equations

KW - Toda lattice

UR - http://www.scopus.com/inward/record.url?scp=84855435891&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84855435891&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84855435891

VL - 10

SP - 267

EP - 305

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

SN - 1539-6746

IS - 1

ER -