Some Remarks About the Two-Matrix Penner Model and the Kazakov-Migdal Model

TL;DR

This paper analyzes the two-matrix Penner model and the Kazakov-Migdal model with logarithmic potentials, providing explicit large-N solutions, connecting to scalar field theories, and calculating gauge field correlators.

Contribution

It offers explicit solutions to the two-matrix and Kazakov-Migdal models with logarithmic potentials using loop equations and Riemann-Hilbert problems, extending understanding of these models.

Findings

Explicit large-N solutions for the models.

Connection to scalar theories in the continuum limit.

Calculation of gauge field pair correlator.

Abstract

I consider the Hermitean 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.