Connecting the Bethe entropy and the edge zeta function of a cycle code

Let the induced Bethe entropy of a pseudo-codeword be the Bethe entropy of the pseudo-marginal that, among all pseudo-marginals that correspond to this pseudo-codeword, has the maximal Bethe entropy. The aim of the present paper is to discuss a connection between the induced Bethe entropy of a cycle code and the edge zeta function of the same code. Namely, we show the equivalence of, on the one hand, the directional derivative of the induced Bethe entropy at the origin in the direction of a pseudo-codeword, and, on the other hand, the growth rate of the coefficients of the monomials that appear in the Taylor series expansion of the edge zeta function and that correspond to this pseudo-codeword. This connection is established via the entropy rate of some random walk that is associated with this pseudo-codeword, this random walk being an object of interest by itself.