Symmetric Gibbs Measures
نویسندگان
چکیده
We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic transformation as exactly the Bernoulli measures—a version of de Finetti’s theorem. Generalizing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and the shift as the Gibbs measures whose potential functions depend on only a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, interval splitting procedures, ‘canonical’ Gibbs states, and the triviality of remote sigma-fields finer than the usual tail field.
منابع مشابه
Entropy of stationary nonequilibrium measures of boundary driven symmetric simple exclusion processes
We examine the entropy of stationary nonequilibrium measures of boundary driven symmetric simple exclusion processes. In contrast with the Gibbs–Shannon entropy [1, 10], the entropy of nonequilibrium stationary states differs from the entropy of local equilibrium states.
متن کاملExchangeable, Gibbs and Equilibrium Measures for Markov Subshifts
We study a class of strongly irreducible, multidimensional, topological Markov shifts, comparing two notions of “symmetric measure”: exchangeability and the Gibbs property. We show that equilibrium measures for such shifts (unique and weak Bernoulli in the one dimensional case) exhibit a variety of spectral properties.
متن کاملAnalytic Properties of Fractional Schrödinger Semigroups and Gibbs Measures for Symmetric Stable Processes
We establish a Feynman-Kac-type formula to define fractional Schrödinger operators for (fractional) Kato-class potentials as self-adjoint operators. In this functional integral representation symmetric α-stable processes appear instead of Brownian motion. We derive asymptotic decay estimates on the ground state for potentials growing at infinity. We prove intrinsic ultracontractivity of the Fey...
متن کاملScaling Limit for a Class of Gradient Fields with Non-convex Potentials
where ̺ is a positive measure with compact support in (0,∞). Hence V is symmetric and nonconvex in general. While for strictly convex V ’s the translation-invariant, ergodic gradient Gibbs measures are completely characterized by their tilt, a non-convex potential as above may lead to several ergodic gradient Gibbs measures with zero tilt. Still, every ergodic, zero-tilt gradient Gibbs measure f...
متن کاملS ep 2 00 4 Gibbs Measures For SOS Models On a Cayley Tree
We consider a nearest-neighbor SOS (solid-on-solid) model, with several spin values 0, 1,. .. , m, m ≥ 2, and zero external field, on a Cayley tree of order k (with k + 1 neighbors). The SOS model can be treated as a natural generalisation of the Ising model (obtained for m = 1). We mainly assume that m = 2 (three spin values) and study translation-invariant (TI) and 'splitting' (S) Gibbs measu...
متن کامل