The broad class of U(N) and SU(N) Polyakov loop models on the lattice
are solved exactly in the combined large N, Nf limit, where N is a number
of colors and Nf is a number of quark flavors, and in any dimension.
In this 't Hooft-Veneziano limit the ratio N/Nf is kept fixed. We calculate
both the free energy and various correlation functions. The critical behavior
of the models is described in details at finite temperatures and non-zero baryon
chemical potential. Furthermore, we prove that the calculation of the N-point
(baryon) correlation function reduces to the geometric median problem in
the confinement phase. In the deconfinement phase we establish an existence
of the complex masses and an oscillating decay of correlations in a certain
region of parameters.