Asymptotic pressure on Cayley graphs of finitely generated semigroups

The vertices of the Cayley graph of a finitely generated semigroup form a set of sites which can be labeled by elements of a finite alphabet in a manner governed by a nonnegative real interaction matrix, respecting nearest neighbor adjacency restrictions. To the set of these configurations there is associated a pressure, which is defined as the limit of averages over certain finite subgraphs of the logarithm of the partition function. We consider families of such systems and prove that the limit of the values of the pressure as the number of generators grows without bound, which we call the asymptotic pressure, equals the logarithm of the maximum row sum of the interaction matrix.

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