Jeffrey E. Steif

Non-uniqueness of measures of maximal entropy for subshifts of finite type

It is known that in one dimension an irreducible subshift of finite type has a unique measure of maximal entropy, the so-called Parry measure. Here we give a counterexample to this in higher dimensions. For this example, we also describe the geometric structure of the measures of maximal entropy and show that there are exactly two extremal measures.

Non-interactive correlation distillation, inhomogeneous Markov chains, and the reverse Bonami-Beckner inequality

AbstractIn this paper we studynon-interactive correlation distillation (NICD), a generalization of noise sensitivity previously considered in [5, 31, 39]. We extend the model toNICD on trees. In this model there is a fixed undirected tree with players at some of the nodes. One node is given a uniformly random string and this string is distributed throughout the network, with the edges of the tree acting as independent binary symmetric channels. The goal of the players is to agree on a shared random bit without communicating.Our new contributions include the following:• In the case of ak-leaf star graph (the model considered in [31]), we resolve the open question of whether the success probability must go to zero ask » ∞. We show that this is indeed the case and provide matching upper and lower bounds on the asymptotically optimal rate (a slowly-decaying polynomial).• In the case of thek-vertex path graph, we show that it is always optimal for all players to use the same 1-bit function.• In the general case we show that all players should use monotone functions. We also show, somewhat surprisingly, that for certain trees it is better if not all players use the same function. Our techniques include the use of thereverse Bonami-Beckner inequality. Although the usual Bonami-Beckner has been frequently used before, its reverse counterpart seems not to be well known. To demonstrate its strength, we use it to prove a new isoperimetric inequality for the discrete cube and a new result on the mixing of short random walks on the cube. Another tool that we need is a tight bound on the probability that a Markov chain stays inside certain sets; we prove a new theorem generalizing and strengthening previous such bounds [2, 3, 6]. On the probabilistic side, we use the “reflection principle” and the FKG and related inequalities in order to study the problem on general trees.

Stationary determinantal processes: Phase multiplicity, Bernoullicity, entropy, and domination

We study a class of stationary processes indexed by

A characterization of competition graphs of arbitrary digraphs

\Z^d

Amenability and Phase Transition in the Ising Model

that are defined via minors of

New results on measures of maximal entropy

d

A survey on dynamical percolation

-dimensional (multilevel) Toeplitz matrices. We obtain necessary and sufficient conditions for phase multiplicity (the existence of a phase transition) analogous to that which occurs in statistical mechanics. Phase uniqueness is equivalent to the presence of a strong

Some Rigorous Results for the Greenberg-Hastings Model

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Random walk in random scenery: A survey of some recent results

property, a particular strengthening of the usual

Propp–Wilson Algorithms and Finitary Codings for High Noise Markov Random Fields

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