A statistical mechanics approach for a rigidity problem

We focus the problem of establishing when a statistical mechanics system is determined by its free energy. A lattice system, modelled by a directed and weighted graph G (whose vertices are the spins and its adjacency matrix M will be given by the system transition rules), is considered. For a matrix A(q), depending on the system interactions, with entries which are in the ring Z[a(q) : a is an element of R+] and such that A(0) equals the integral matrix M, the system free energy beta(A)(q) will be defined as the spectral radius of A(q). This kind of free energy will be related with that normally introduced in Statistical Mechanics as proportional to the logarithm of the partition function. Then we analyze under what conditions the following statement could be valid: if two systems have respectively matrices A,B and beta(A) = beta(B) then the matrices are equivalent in some sense. Issues of this nature receive the name of rigidity problems. Our scheme, for finite interactions, closely follows that developed, within a dynamical context, by Pollicott and Weiss but now emphasizing their statistical mechanics aspects and including a classification for Gibbs states associated to matrices A(q). Since this procedure is not applicable for infinite range interactions, we discuss a way to obtain also some rigidity results for long range potentials.