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I investigate the Kazakov-Migdal (KM) model -- the Hermitean gauge-invariantmatrix model on a D-dimensional lattice. I utilize an exact large-N solution ofthe KM model with a logarithmic potential to examine its critical behavior. Ifind critical lines associated with gamma_{string}=-1/2 and gamma_{string}=0 aswell as a tri-critical point associated with a Kosterlitz-Thouless phasetransition. The continuum theories are constructed expanding around thecritical points. The one associated with gamma_{string}=0 coincides with thestandard d=1 string while the Kosterlitz-Thouless phase transition separates itfrom that with gamma_{string}=-1/2 which is indistinguishable from pure 2Dgravity for local observables but has a continuum limit for correlators ofextended Wilson loops at large distances due to a singular behavior of theItzykson-Zuber correlator of the gauge fields. I reexamine the KM model with anarbitrary potential in the large-D limit and show that it reduces at large N toa one-matrix model whose potential is determined self-consistently. A relationwith discretized random surfaces is established via the gauged Potts modelwhich is equivalent to the KM model at large N providing the coordinationnumbers coincide.Comment: 45pp., Latex, YM-4-9

CRITICAL SCALING AND CONTINUUM LIMITS IN THE D>1 KAZAKOV-MIGDAL MODEL