Critical Scaling and Continuum Limits in the D>1 Kazakov-Migdal Model

I investigate the Kazakov-Migdal (KM) model -- the Hermitean gauge-invariant matrix model on a D-dimensional lattice. I utilize an exact large-N solution of the KM model with a logarithmic potential to examine its critical behavior. I find critical lines associated with gamma_{string}=-1/2 and gamma_{string}=0 as well as a tri-critical point associated with a Kosterlitz-Thouless phase transition. The continuum theories are constructed expanding around the critical points. The one associated with gamma_{string}=0 coincides with the standard d=1 string while the Kosterlitz-Thouless phase transition separates it from that with gamma_{string}=-1/2 which is indistinguishable from pure 2D gravity for local observables but has a continuum limit for correlators of extended Wilson loops at large distances due to a singular behavior of the Itzykson-Zuber correlator of the gauge fields. I reexamine the KM model with an arbitrary potential in the large-D limit and show that it reduces at large N to a one-matrix model whose potential is determined self-consistently. A relation with discretized random surfaces is established via the gauged Potts model which is equivalent to the KM model at large N providing the coordination numbers coincide.