Some remarks about the two-matrix Penner model and the Kazakov-Migdal model

Abstract

I consider the hermitian two-matrix model with a logarithmic potential which is associated in the one-matrix case with the Penner model. Using loop equations I find an explicit solution of the model at large N (or in the spherical approximation) and demonstrate that it solves the corresponding Riemann-Hilbert problem. I construct the potential of the Kazakov-Migdal model on a D-dimensional lattice, which turns out to be a sum of two logarithms as well, whose large N solution is given by the same formulas. In the “naive” continuum limit this potential recovers in D > 4 dimensions the standard scalar theory with quartic self-interaction. I exploit the solution to calculate explicitly the pair correlator of gauge fields in the Kazakov-Migdal model with the logarithmic potential.