In this paper, we consider an extended Kazakov-Migdal model defined on an
    arbitrary graph. The partition function of the model, which is expressed as the
    summation of all Wilson loops on the graph, turns out to be represented by the
    Bartholdi zeta function weighted by unitary matrices on the edges of the graph.
    The partition function on the cycle graph at finite $N$ is expressed by the
    generating function of the generalized Catalan numbers. The partition function
    on an arbitrary graph can be exactly evaluated at large $N$ which is expressed
    as an infinite product of a kind of deformed Ihara zeta function. The non-zero
    area Wilson loops do not contribute to the leading part of the $1/N$-expansion
    of the free energy but to the next leading. The semi-circle distribution of the
    eigenvalues of the scalar fields is still an exact solution of the model at
    large $N$ on an arbitrary regular graph, but it reflects only zero-area Wilson

    Source link


    Leave A Reply