arXiv subjects
Publications and source records attributed to Ronnie Pavlov.
For any stationary $\mZ^d$-Gibbs measure that satisfies strong spatial mixing, we obtain sequences of upper and lower approximations that converge to its entropy. In the case, $d=2$, these approximations are efficient in the sense that the approximations are accurate to within $ε$ and can be computed in time polynomial in $1/ε$.
In this paper we show that any one-dimensional stationary, finite-valued Markov Random Field (MRF) is a Markov chain, without any mixing condition or condition on the support. Our proof makes use of two properties of the support $X$ of a finite-valued stationary MRF: 1) $X$ is non-wandering (this is a property of the support of any finite-valued stationary process) and 2) $X$ is a topological Markov field (TMF). The latter is a new property that sits in between the classes of shifts of finite type and sofic shifts, which are well-known objects of study in symbolic dynamics. Here, we develop the TMF property in one dimension, and we will develop this property in higher dimensions in a future paper. While we are mainly interested in discrete-time finite-valued stationary MRF's, we also consider continuous-time, finite-valued stationary MRF's, and show that these are (continuous-time) Markov chains as well.
For any two-dimensional nearest neighbor shift of finite type X and any integer n > 0, one can define the horizontal strip shift H_n(X) to be the set of configurations on Z x {1,...,n} which do not contain any forbidden transitions for X. It is always the case that the sequence h(H_n(X))/n of normalized topological entropies of the strip shifts approaches h(X), the topological entropy of X. In this paper, we use probabilistic methods from interacting particle systems to show that for the two-dimensional hard square shift H, in fact h(H_{n+1}(H)) - h(H_n(H)) also approaches h(H), and the rate of convergence is at least exponential. A consequence of this is that h(H) is computable to any tolerance 1/n in time polynomial in n. We also give an example of a two-dimensional block gluing nearest neighbor shift of finite type Y for which h(H_{n+1}(Y)) - h(H_n(Y)) does not even approach a limit.
For a class of $\zz^2$ Markov Random Fields (MRFs) $μ$, we show that the sequence of successive differences of entropies of induced MRFs on strips of height $n$ converges exponentially fast (in $n$) to the entropy of $μ$. These strip entropies can be computed explicitly when $μ$ is a Gibbs state given by a nearest-neighbor interaction on a strongly irreducible nearest-neighbor $\zz^2$ shift of finite type $X$. We state this result in terms of approximations to the (topological) pressures of certain functions on such an $X$, and we show that these pressures are computable if the values taken on by the functions are computable. Finally, we show that our results apply to the hard core model and Ising model for certain parameter values of the corresponding interactions, as well as to the topological entropy of certain nearest-neighbor $\zz^2$ shifts of finite type, generalizing a result in \cite{Pa}.