The Infinite-State Potts Model and Solid Partitions of an Integer

It has been established that the infinite-state Potts model in d dimensions generates restricted partitions of integers in d-1 dimensions, the latter a well-known intractable problem in number theory for d>3. Here we consider the d=4 problem. We consider a Potts model on an L × M × N × P hypercubic lattice whose partition function G_LMNP(t) generates restricted solid partitions on an L × M × N lattice with each part no greater than P. Closed-form expressions are obtained for G_222P(t) and we evaluated its zeroes in the complex t plane for different values of P. On the basis of our numerical results we conjecture that all zeroes of the enumeration generating function G_LMNP(t) lie on the unit circle |t|=1 in the limit that any of the indices L, M, N, P becomes infinite.