Published: Sept. 24, 2013

Continuum Limits of the Toda Lattice for Map Enumeration

Virgil Pierce

Department of MathematicsUniversity of Texas - Pan American

Date and time: 

Tuesday, September 24, 2013 - 4:00pm

Location: 

ECOT 226

Abstract: 

We are considering a family of N-by-N Hermitian random matrices whose entries are Gaussian normal random variables.  The partition function of this ensemble, which generates the expected values of symmetric polynomials in the eigenvalues, is an interesting mathematical construction.  The log-partition function possess an asymptotic expansion in inverse powers of the size parameter N whose terms are generating functions for the enumeration of maps (or ribbon graphs) partitioned by the genus and indexed by the degree of the vertices.  The partition function is also a tau function of the Toda lattice hierarchy, a classic example of an integrable system.  The asymptotic expansion of the partition function then induces a continuum limit on the Toda hierarchy which ultimately can be used as a system of equations determining the generating functions.  We give explicit formulas for the generating functions in at least some cases, thus solving the combinatoric problem.  We will discuss the difficulties encountered in extending these results to more complicated families of maps.