Gross-Witten-Wadia Phase Transition in Induced QCD on the Graph

In this paper, we examine a modification of the Kazakov-Migdal (KM) model with gauge group $U(N_c)$, where the adjoint scalar fields in the conventional KM model are replaced by $N_f$ fundamental scalar fields (FKM model). After tuning the coupling constants and eliminating the fundamental scalar fields, the partition function of this model is expressed as an integral of a graph zeta function weighted by unitary matrices. The FKM model on cycle graphs at large $N_c$ 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 model on a general graph in two distinct parameter regimes and demonstrate that the FKM model generally consists of multiple phases. The effective action of the FKM model reduces to the standard Wilson action by taking an appropriate scaling limit when the graph consists of plaquettes (fundamental cycles) of the same size, as in the square lattice case. We show that, for the FKM model on such a graph, the third-order GWW phase transition universally occurs in this scaling limit.