Combinatorics of the interaction on plane lattices

We consider the problem on enumerating the events appearing on the edges of regular plane lattices under given interactions at the nodes. We investigate a simple case where the interactions are described by a $(0,1)$-matrix of size $4\times 4$. In particular, we study the asymptotic behaviour of the number of events as the linear sizes of the lattice tend to infinity and give estimates of the exponential growth of this number as functions of the matrix describing the interactions.
This research was supported by the Russian Foundation for Basic Research, grant 97-01-00627, and by INTAS, grant 94-0469.