Ronnie Pavlov

Approximating the hard square entropy constant with probabilistic methods

For any two-dimensional nearest neighbor shift of finite type X and any integer n≥1, one can define the horizontal strip shift Hn(X) to be the set of configurations on Z×{1,…,n} which do not contain any forbidden pairs of adjacent letters for X. It is always the case that the sequence htop(Hn(X))/n of normalized topological entropies of the strip shifts converges to htop(X), the topological entropy of X. In this paper, we combine ergodic theoretic techniques with methods from percolation theory and interacting particle systems to show that for the two-dimensional hard square shift H, the sequence htop(Hn+1(H))−htop(Hn(H)) also converges to htop(H), and that the rate of convergence is at least exponential. As a corollary, we show that htop(H) is computable to any tolerance e in time polynomial in 1/e. We also show that this phenomenon is not true in general by defining a block gluing two-dimensional nearest neighbor shift of finite type Y for which htop(Hn+1(Y))−htop(Hn(Y)) does not even approach a limit.

Classification of sofic projective subdynamics of multidimensional shifts of finite type

Motivated by Hochman’s notion of subdynamics of a Z subshift [8], we define and examine the projective subdynamics of Z shifts of finite type (SFTs) where we restrict not only the action but also the phase space. We show that any Z sofic shift of positive entropy is the projective subdynamics of a Z2 (Z) SFT, and that there is a simple condition characterizing the class of zero-entropy Z sofic shifts which are not the projective subdynamics of any Z2 SFT. We define notions of stable and unstable subdynamics in analogy with the notions of stable and unstable limit sets in cellular automata theory, and discuss how our results fit into this framework. One-dimensional strictly sofic shifts of positive entropy admit both a stable and an unstable realization, whereas a particular class of zero-entropy Z sofics only allows for an unstable realization. Finally, we prove that the union of Z subshifts all of which are realizable in Z SFTs is again realizable when it contains at least two periodic points, that the projective subdynamics of Z2 SFTs with the uniform filling property (UFP) are always sofic and we exhibit a class of non-sofic Z subshifts which are not the subdynamics of any Z SFT.

Approximating entropy for a class of ℤ 2 Markov random fields and pressure for a class of functions on ℤ 2 shifts of finite type

For a class of

COMPUTING BOUNDS FOR ENTROPY OF STATIONARY Z d MARKOV RANDOM FIELDS

\zz^2

An integral representation for topological pressure in terms of conditional probabilities

Markov Random Fields (MRFs)

One-dimensional Markov random fields, Markov chains and topological Markov fields

\mu

A Class of Nonsofic Multidimensional Shift Spaces

, we show that the sequence of successive differences of entropies of induced MRFs on strips of height

Perturbations of multidimensional shifts of finite type

n

Representation and Poly-time Approximation for Pressure of \mathbb {Z}^2 Lattice Models in the Non-uniqueness Region

converges exponentially fast (in

Strong Spatial Mixing in Homomorphism Spaces

n