Markov approximations of Gibbs measures for long-range interactions on 1D lattices

Abstract

Abstract. We study one-dimensional lattice systems with pair-wise interactions of infinite range. We show projective convergence of Markov measures to the unique equilibrium state. For this purpose we impose a slightly stronger condition than summability of variations on the regularity of the interaction. With our condition we are able to explicitly obtain stretched exponential bounds for the rate of mixing of the equilibrium state. Finally we show convergence for the entropy of the Markov measures to that of the equilibrium state via the convergence of their topological pressure.

Information Entropy Production of Maximum Entropy Markov Chains from Spike Trains

“…Remark 8: Here we have obtained the transfer matrix and its unique invariant probability measure from a potential of finite range, which is a Markov measure that satisfies the variational principle [3]. Moreover, is the same as the unique Gibbs measure associated to a finite range potential, that takes the form (17) [18]. For finite range potentials H in one dimension, one has always a unique equilibrium measure which satisfies the "Gibbs property" (19).…”

“…In an analogous way, as it is done for Markov approximations of Gibbs measures [ 40 , 41 ], for a finite range potential , we introduce the transfer matrix , …”

“…N , in this case we write σ 0,L−1 ∼ σ ( ) , σ ( ) . In an analogous way as done for Markov approximations of Gibbs measures [3,18], we introduce here the transfer matrix L, which in some sense is a generalization of the concept of Markov transition matrix, as follows:…”

Projective distance and $g$-measures

“…Furthermore, since in that case the relative entropy of the limiting Gibbs measure with respect to the approximations goes to cero, then Marton's bounds [21,22] ensures the convergence of the approximations in d-distance. In a recent work [20], Maldonado and Salgado applied our approach to study the approximability of Gibbs measure for two-body interactions in one dimensional symbolic systems. This technique was also used in our study of the preservation of Gibbsianness under amalgamation of symbols [6].…”