Kazakov–Migdal model with logarithmic potential and the double Penner matrix model

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→∞ 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 an explicitly soluble model. The formal solution to this double Penner KM Model is reviewed first. Special attention is paid to the relationship of this model to an ordinary (one) matrix model whose potential has two logarithmic singularities (the double Penner model). A detailed analysis is presented of the large N behavior of this double Penner model. The various one cut and two cut solutions are described and cases in which ‘‘eigenvalue condensation’’ occurs at the singular points of the potential are discussed. Then the consequences of our study for the KM model described above are discussed. The phase diagram of the model is presented and its critical regions are described.