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In this paper, we examine a modification of the Kazakov-Migdal (KM) modelwith gauge group $U(N_c)$, where the adjoint scalar fields in the conventionalKM model are replaced by $N_f$ fundamental scalar fields (FKM model). Aftertuning the coupling constants and eliminating the fundamental scalar fields,the partition function of this model is expressed as an integral of a graphzeta function weighted by unitary matrices. The FKM model on cycle graphs atlarge $N_c$ exhibits the Gross-Witten-Wadia (GWW) phase transition only when$N_f > N_c$. In the large $N_c$ limit, we evaluate the free energy of the modelon a general graph in two distinct parameter regimes and demonstrate that theFKM model generally consists of multiple phases. The effective action of theFKM model reduces to the standard Wilson action by taking an appropriatescaling limit when the graph consists of plaquettes (fundamental cycles) of thesame size, as in the square lattice case. We show that, for the FKM model onsuch a graph, the third-order GWW phase transition universally occurs in thisscaling limit.

GWW Phase Transition in Induced QCD on the Graph