Predictability, topological entropy, and invariant random orders

Alpeev, Andrei; Meyerovitch, Tom; Ryu, Sieye

doi:10.1090/proc/15158

Cited by 4 publications

(12 citation statements)

References 24 publications

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“…In Section 3, we show that for non-amenable groups it not always possible to extend a left-invariant partial order to a left-invariant random (total) order. This answers a question posed in [2], where extendability of partial invariant random orders in the amenable case was resolved. More specifically, we demonstrate non-extendability as above with respect to the left-invariant partial order on the group SL 3 (Z) corresponding to the semi-group of matrices that preserve the positive octant.…”

Section: Introductionmentioning

confidence: 62%

“…Proposition 2.4 ([23], [2]). Any invariant random partial order on an amenable group can be extended to an invariant random (total) order.…”

Section: Definitions and Basic Observationsmentioning

confidence: 99%

“…A proof of the above proposition follows by observing that the space of (not necessarily invariant) extensions of a given random order is a non-empty simplex on which the group acts. See [2] for details.…”

Section: Definitions and Basic Observationsmentioning

confidence: 99%

“…The central objects of this paper are invariant random orders, namely probability measures on the space of orders whose distribution is invariant with respect to multiplication (say, from the left). The term "invariant random orders" appeared in [2], and we refer the reader to [13,23] for earlier applications, in particular in the context of entropy theory for actions of amenable groups. There are further recent applications of invariant random orders in the context of entropy theory [4,8].…”

Section: Introductionmentioning

confidence: 99%

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Extensions of invariant random orders on groups

Glasner¹,

Lin²,

Meyerovitch³

2022

Preprint

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In this paper we study the action of a countable group Γ on the space of orders on the group. In particular, we are concerned with the invariant probability measures on this space, known as invariant random orders. We show that for any countable group the space of random invariant orders is rich enough to contain an isomorphic copy of any free ergodic action, and characterize the non-free actions realizable. We prove a Glasner-Weiss dichotomy regarding the simplex of invariant random orders. We also show that the invariant partial order on SL 3 (Z) corresponding to the semigroup of positive matrices cannot be extended to an invariant random total order. We thus provide the first example for a partial order (deterministic or random) that cannot be randomly extended.

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Section: Introductionmentioning

confidence: 62%

“…Proposition 2.4 ([23], [2]). Any invariant random partial order on an amenable group can be extended to an invariant random (total) order.…”

Section: Definitions and Basic Observationsmentioning

confidence: 99%

Section: Definitions and Basic Observationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Extensions of invariant random orders on groups

Glasner¹,

Lin²,

Meyerovitch³

2022

Preprint

Self Cite

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“…Equivalently, is a subsemigroup of with (we remark that left-invariant partial orders on groups also appeared in [AMR21] but for different reasons). A subset of satisfying the above two axioms is called a positive semigroup .…”

Section: Introductionmentioning

confidence: 99%

Harmonic models and Bernoullicity

Hayes¹

2021

Compositio Math.

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We give many examples of algebraic actions which are factors of Bernoulli shifts. These include certain harmonic models over left-orderable groups of large enough growth, as well as algebraic actions associated to certain lopsided elements in any left-orderable group. For many of our examples, the acting group is amenable so these actions are Bernoulli (and not just a factor of a Bernoulli), but there is no obvious Bernoulli partition.

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Counting independent sets in amenable groups

Briceño

2023

Ergod. Th. Dynam. Sys.

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Given a locally finite graph $\Gamma $ , an amenable subgroup G of graph automorphisms acting freely and almost transitively on its vertices, and a G-invariant activity function $\unicode{x3bb} $ , consider the free energy $f_G(\Gamma ,\unicode{x3bb} )$ of the hardcore model defined on the set of independent sets in $\Gamma $ weighted by $\unicode{x3bb} $ . Under the assumption that G is finitely generated and its word problem can be solved in exponential time, we define suitable ensembles of hardcore models and prove the following: if $\|\unicode{x3bb} \|_\infty < \unicode{x3bb} _c(\Delta )$ , there exists a randomized $\epsilon $ -additive approximation scheme for $f_G(\Gamma ,\unicode{x3bb} )$ that runs in time $\mathrm {poly}((1+\epsilon ^{-1})\lvert \Gamma /G \rvert )$ , where $\unicode{x3bb} _c(\Delta )$ denotes the critical activity on the $\Delta $ -regular tree. In addition, if G has a finite index linearly ordered subgroup such that its algebraic past can be decided in exponential time, we show that the algorithm can be chosen to be deterministic. However, we observe that if $\|\unicode{x3bb} \|_\infty> \unicode{x3bb} _c(\Delta )$ , there is no efficient approximation scheme, unless $\mathrm {NP} = \mathrm {RP}$ . This recovers the computational phase transition for the partition function of the hardcore model on finite graphs and provides an extension to the infinite setting. As an application in symbolic dynamics, we use these results to develop efficient approximation algorithms for the topological entropy of subshifts of finite type with enough safe symbols, we obtain a representation formula of pressure in terms of random trees of self-avoiding walks, and we provide new conditions for the uniqueness of the measure of maximal entropy based on the connective constant of a particular associated graph.

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Predictability, topological entropy, and invariant random orders

Cited by 4 publications

References 24 publications

Extensions of invariant random orders on groups

Extensions of invariant random orders on groups

Harmonic models and Bernoullicity

Counting independent sets in amenable groups

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