On asymptotic properties of equilibrium measures corresponding to finite submatrices of infinite nonnegative matrices

Abstract

There is a canonical way to assign a translation invariant Markov measure to every finite irreducible nonnegative matrix. This measure is defined on a sequence space, translation invariant, and satisfies a variational principle (due to the last property it is said to be equilibrium). The same can be done for some infinite nonnegative matrices. If A is such a matrix, one can consider an increasing sequence of its finite irreducible submatrices A _n that tends to A in a natural sense, and ask if the equilibrium measure μ _An assigned to A _n converges to the equilibrium measure μ _A assigned to A. For two classes of matrices A, the answer does not depend on {A _n }: for one of these classes, μ _An → μ _A , for the other one, μ _An converges to the zero measure. We describe, in geometric terms, a third class located ‘between’ the above two for which the situation is also intermediate: for some sequences {A _n } the asymptotic behavior of μ _An is as in the first class, while for some other sequences the behavior is as in the second one.