Generalized Kontsevich model versus Toda hierarchy and discrete matrix models

Abstract

We present 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-time” and “zero-time” variables: they appear to modify the original GKM action by negative-power and logarithmic contributions, respectively. It is shown that such a deformed τ-function satisfies the same string equation as the original one. In the case of quardratic potentials GKM turns out to describe forced Toda chain hierarchy and thus corresponds to a discrete matrix model, with the role of matrix size played by the zero-time (at integer positive points). This relation allows one to discuss the double-scaling continuum limit entirely in terms of GKM, i.e. essentially in terms of finite-fold integrals.