Generalized Kontsevich model versus Toda hierarchy and discrete matrix models

Abstract

We represent the partition function of the Generalized Kontsevich Model (GKM) in the form of a Toda lattice τ -function and discuss various implications of non-vanishing "negative"-and "zero"-time variables: the appear to modify the original GKM action by negative-power and logarithmic contributions respectively. It is shown that so deformed τ -function satisfies the same string equation as the original one. In the case of quadratic potential GKM turns out to describe forced Toda chain hierarchy and, thus, corresponds to a discrete matrix model, with the role of the matrix size played by the zero-time (at integer positive points). This relation allows one to discuss the doublescaling continuum limit entirely in terms of GKM, i.e. essentially in terms of finite-fold integrals.

Refined open intersection numbers and the Kontsevich-Penner matrix model

“…String and dilaton equations for the Kontsevich-Penner model were derived in [2,5] (In a more general setup of the Generalized Kontsevich Model the string equation in terms of the eigenvalues of the external matrix was derived already in [12]). They coincide with the equations for the extended refined open partition function, derived in sections 3.5 and 3.6. , n ≥ 1, a 1 , .…”

Topological expansion of the 2-matrix model correlation functions: diagrammatic rules for a residue formula

“…planar topologies), the solution of loop equations is known to be related to Toda hierarchy [9,28,35,36]. For this reason, the large N expansion of matrix models plays an important role in integrable systems, and in many areas of physics [29].…”

Canonical and Grand Canonical Partition Functions of Dyson Gases as Tau-Functions of Integrable Hierarchies and Their Fermionic Realization

“…Similar constructions in the context of random matrix models of different types were given in [3,4,5,6]. However, most of the previous studies were devoted to canonical ensembles with a fixed number of particles, N, which after F. Dyson [7] are customarily viewed as eigenvalues of a N × N random matrix.…”