Christopher Shriver
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View article: ALGEBRAIC DYNAMICAL SYSTEMS FROM LDPC CODES SATISFY A STRONG NEGATION OF THE WEAK PINSKER PROPERTY
ALGEBRAIC DYNAMICAL SYSTEMS FROM LDPC CODES SATISFY A STRONG NEGATION OF THE WEAK PINSKER PROPERTY Open
We construct an explicit algebraic example of a subshift of finite type over a group $\Gamma $ with an invariant Markov measure which has completely positive sofic entropy (with respect to ‘most’ sofic approximations) and yet does not have…
View article: Equilibrium and nonequilibrium Gibbs states on sofic groups
Equilibrium and nonequilibrium Gibbs states on sofic groups Open
Recent work of Barbieri and Meyerovitch has shown that, for very general spin systems indexed by sofic groups, equilibrium (i.e. pressure-maximizing) states are Gibbs. The main goal of this paper is to show that the converse fails in an in…
View article: Algebraic dynamical systems from LDPC codes satisfy a strong negation of the weak Pinsker property
Algebraic dynamical systems from LDPC codes satisfy a strong negation of the weak Pinsker property Open
We construct an explicit algebraic example of a subshift of finite type over a group $Γ$ with an invariant Markov measure which has completely positive sofic entropy (with respect to `most' sofic approximations) and yet does not have a dir…
View article: Typical sofic entropy and local limits for free group shift systems
Typical sofic entropy and local limits for free group shift systems Open
We show that for any invariant measure $μ$ on a free group shift system, there are two numbers $h^\flat \leq h^\sharp$ which in some sense are the typical upper and lower sofic entropy values. We also give a condition under which $h^\flat …
View article: Equilibrium and nonequilibrium Gibbs states on sofic groups
Equilibrium and nonequilibrium Gibbs states on sofic groups Open
Recent work of Barbieri and Meyerovitch has shown that, for very general spin systems indexed by sofic groups, equilibrium (i.e. pressure-maximizing) states are Gibbs. The main goal of this paper is to show that the converse fails in an in…
View article: The relative <i>f</i>-invariant and non-uniform random sofic approximations
The relative <i>f</i>-invariant and non-uniform random sofic approximations Open
The f -invariant is an isomorphism invariant of free-group measure-preserving actions introduced by Lewis Bowen, who first used it to show that two finite-entropy Bernoulli shifts over a finitely generated free group can be isomorphic only…
View article: Metastability and maximal-entropy joinings of Gibbs measures on finitely-generated groups
Metastability and maximal-entropy joinings of Gibbs measures on finitely-generated groups Open
We prove a metastability result for finitary microstates which are good models for a Gibbs measure for a nearest-neighbor interaction on a finitely-generated group. This is used to show that any maximal-entropy joining of two such Gibbs st…
View article: Free Energy, Gibbs Measures, and Glauber Dynamics for Nearest-neighbor Interactions on Trees
Free Energy, Gibbs Measures, and Glauber Dynamics for Nearest-neighbor Interactions on Trees Open
We extend results of R. Holley beyond the integer lattice to a large class of groups which includes free groups. In particular we show that a shift-invariant measure is Gibbs if and only if it is Glauber-invariant. Moreover, any shift-inva…
View article: The relative f-invariant and non-uniform random sofic approximations
The relative f-invariant and non-uniform random sofic approximations Open
The $f$-invariant is an isomorphism invariant of free-group measure-preserving actions introduced by Lewis Bowen in [arXiv:0802.4294], where it was used to show that two finite-entropy Bernoulli shifts over a finitely generated free group …
View article: Concentration of Broadcast Models on Trees
Concentration of Broadcast Models on Trees Open
An inequality of K. Marton shows that the joint distribution of a Markov chain with uniformly contracting transition kernels exhibits concentration. We prove an analogous inequality for broadcast models on finite trees. We use this inequal…
View article: An Application of Markov Chain Analysis to Integer Complexity
An Application of Markov Chain Analysis to Integer Complexity Open
The complexity $f(n)$ of an integer was introduced in 1953 by Mahler & Popken: it is defined as the smallest number of $1$'s needed in conjunction with arbitrarily many +, * and parentheses to write an integer $n$ (for example, $f(6) \leq …