Complete equivalence of Gibbs ensembles for one-dimensional Markov systems (Q2707198)

We show the equivalence of the Gibbs ensembles at the level of measures for one-dimensional Markov systems with arbitrary boundary conditions. That is, the limit of the microcanonical Gibbs ensemble is a Gibbs measure with an interaction depending on the microcanonical constraint. In fact the usual microcanonical condition is replaced by the sharper constraint that all type frequencies of neighboring spins (including the boundary spins) are fixed. When conditioning on a set of different frequencies of neighboring spins compatible with physical quantities like energy density we get the usual microcanonical ensemble. We show that the limit is a Gibbs measure for a nearest neighbor potential depending on the pair measure which maximizes the entropy on the given set of pair measures. For this we show the large deviation property of the pair empirical measure for arbitrary boundary conditions [see \textit{J.-D. Deuschel}, \textit{D. W. Stroock} and \textit{H. Zessin}, Commun. Math. Phys. 139, No. 1, 83-101 (1991; Zbl 0727.60025) and \textit{H.-O. Georgii}, Ann. Probab. 21, No. 4, 1845-1875 (1993; Zbl 0790.60031) for periodic boundary condition]. We establish analogous results for finite range potentials [see the author, J. Stat. Mech. 105, No. 516, 879-908 (2001)]. In the second part of the thesis we show the same result for the one-dimensional Ising model, where we use the combinatoric approach via the \(n\)-particle Fibonacci numbers.