Kazakov--Migdal Model with Logarithmic Potential and the Double Penner Matrix Model

TL;DR

This paper analyzes a solvable variant of the Kazakov--Migdal lattice gauge theory with a logarithmic potential, exploring its large N behavior, phase structure, and eigenvalue condensation phenomena.

Contribution

It provides an explicit solution to the Double Penner Kazakov--Migdal model and examines its phase diagram and critical behavior.

Findings

Explicit large N solutions for the Double Penner model

Identification of one-cut and two-cut eigenvalue solutions

Description of phase transitions and eigenvalue condensation

Abstract

The Kazakov--Migdal (KM) Model is a U(N) Lattice Gauge Theory with a Scalar Field in the adjoint representation but with no kinetic term for the Gauge Field. This model is formally soluble in the limit $N\rightarrow \infty$ though explicit solutions are available for a very limited number of scalar potentials. A ``Double Penner'' Model in which the potential has two logarithmic singularities provides an example of a explicitly soluble model. We begin by reviewing the formal solution to this Double Penner KM Model. We pay special attention to the relationship of this model to an ordinary (one) matrix model whose potential has two logarithmic singularities (the Double Penner Model). We present a detailed analysis of the large N behavior of this Double Penner Model. We describe the various one cut and two cut solutions and we discuss cases in which ``eigenvalue condensation'' occurs at the singular points of the potential. We then describe the consequences of our study for the KM Model described above. We present the phase diagram of the model and describe its critical regions.