Random ordering formula for sofic and Rokhlin entropy of Gibbs measures

Abstract

We prove the explicit formula for sofic and Rokhlin entropy of actions arising from some class of Gibbs measures. It provides a new set of examples with sofic entropy independent of sofic approximations. It is particularilly interresting, since in non-amenable case Rokhlin entropy was computed only in case of Bernoulli actions and for some examples with zero Rokhlin entropy. As an example we show that our formula holds for the supercritical Ising model. We also establish a criterion for uniqueness of Gibbs measure by means of f -invariant.

Predictability, topological entropy and invariant random orders

“…See the book [8] for background and much more. Particular invariant random orders have been applied to entropy theory, notably Keiffer's paper [21] about actions of amenable groups, and [1,3,31] for actions of countable but not necessarily amenable groups.…”

Entropy, products, and bounded orbit equivalence

“…One property shared by the groups G in this theorem is that their first 2 -Betti number vanishes, which in the non-amenable world can be roughly intuited as an expression of anti-freeness, and indeed our approach breaks down for free groups (see [25, §3.5]). In what is surely not a coincidence, groups whose Bernoulli actions are cocycle superrigid also have vanishing first 2 -Betti number [35], and it has been speculated that these two properties are equivalent in the non-amenable realm (curiously, however, Bernoulli cocycle or orbit equivalence superrigidity remains generally unknown for wreath products of the form Z H with H non-amenable, which satisfy the hypotheses on G above).…”

“…Given that non-amenable products of countably infinite groups form a standard class of examples within the circle of ideas around superrigidity, cost one, and vanishing first 2 -Betti number, and in particular are known to satisfy Bernoulli cocycle superrigidity by a theorem of Popa [36], it is natural to wonder whether the entropy inequality (1) holds if G is instead assumed to be such a product. In [25] we demonstrated, in analogy with Gaboriau's result on cost for products of equivalence relations [13], that product actions of non-locally-finite product groups, when equipped with an arbitrary invariant probability measure, satisfy property SC, which is sufficient for establishing (1).…”

“…. , n} G equipped with a Gibbs measure satisfying one of various uniqueness conditions [1,2]. For the definition of w-normality see the paragraph before Theorem 3.12.…”