Matrix Model Partition Functions Research Articles

Scalar Products of Symmetric Functions and Matrix Integrals

We present relations between Hirota-type bilinear operators, scalar products on spaces of symmetric functions and integrals defining matrix model partition functions. Using the fermionic Fock space representation, a proof of the expansion of an associated class of KP and 2-Toda tau functions $\tau_{r,n}$ in a series of Schur functions generalizing the hypergeometric series is given and related to the scalar product formulae. It is shown how special cases of such $\tau$-functions may be identified as formal series expansions of partition functions. A closed form exapnsion of $\log \tau_{r,n}$ in terms of Schur functions is derived.

Factorisations for partition functions of random Hermitian matrix models

The partition functionZ N , for Hermitian-complex matrix models can be expressed as an explicit integral over ℝ N , whereN is a positive integer. Such an integral also occurs in connexion with random surfaces and models of two dimensional quantum gravity. We show thatZ N can be expressed as the product of two partition functions, evaluated at translated arguments, for another model, giving an explicit connexion between the two models. We also give an alternative computation of the partition function for theφ 4-model. The approach is an algebraic one and holds for the functions regarded as formal power series in the appropriate ring.