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Entropy of Nonamenable Group Actions.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Entropy of Nonamenable Group Actions./
作者:	Shriver, Christopher Edgar.
出版者:	Ann Arbor : ProQuest Dissertations & Theses, : 2021,
面頁冊數:	148 p.
附註:	Source: Dissertations Abstracts International, Volume: 82-12, Section: B.
Contained By:	Dissertations Abstracts International82-12B.
標題:	Theoretical mathematics. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28543597
ISBN:	9798516064456

Entropy of Nonamenable Group Actions.
Shriver, Christopher Edgar.

Entropy of Nonamenable Group Actions. - Ann Arbor : ProQuest Dissertations & Theses, 2021 - 148 p.

Source: Dissertations Abstracts International, Volume: 82-12, Section: B.

Thesis (Ph.D.)--University of California, Los Angeles, 2021.

This item must not be sold to any third party vendors.

Sofic entropy is an isomorphism invariant of measure-preserving actions of sofic groups introduced by Lewis Bowen around 2010. Its classical analogue was introduced in the 1950s by Kolmogorov and Sinai in order to show that Bernoulli shifts over ℤ are nonisomorphic when their base measures have different Shannon entropies. This entropy rate was actively studied over the next few decades and extended to arbitrary amenable groups by Ornstein and Weiss.On the one hand, amenable groups provide an appropriate setting for entropy theory since they have a way of performing the kind of average used to define an entropy rate. On the other hand, statistical physicists have long been interested in some nonamenable structures, such as the Bethe lattice. The problem of finding an appropriate entropy notion for nonamenable group actions, and in particular the problem of isomorphism of Bernoulli shifts in this setting, remained open until Bowen's work. One way to briefly summarize the idea of sofic entropy is to say that we consider the entropy per site along a sequence of large finite systems which locally approximate the infinite one (called a sofic approximation) rather than large finite subsystems of the infinite one. An interesting problem which arises is what effect the choice of sofic approximation can have on the sofic entropy rate.This thesis presents related work on several different problems of sofic entropy theory. In Chapter 2 we study the f-invariant, a variant of sofic entropy for free-group actions introduced by Bowen which can be defined using a kind of uniform random sofic approximation. We use a relative version of the f-invariant to show that the sofic entropy over a kind of "stochastic block model'' random sofic approximation is given by the solution to an entropy-maximization problem. Understanding this optimization problem may shed further light on the dependence of sofic entropy on the sofic approximation.Chapter 4 uses a new notion of sofic free energy density to study Gibbs measures and Glauber dynamics for nearest-neighbor interacting particle systems on some nonamenable groups. The main results are that, under certain reasonable conditions, every Glauber-invariant, shift-invariant measure is Gibbs and that the Glauber evolution of any shift-invariant measure converges to the set of Gibbs measures. These extend results of Holley (1971) for the Ising model on integer lattices.Chapter 5 begins by proving a metastability result for states on finite graphs which are locally similar to the Cayley graph of a finitely-generated group: the Glauber evolution of any state on a finite graph will converge to the unique Gibbs state, but we show that if the initial state is "pseudo-Gibbs'' in that it is in some sense consistent with some Gibbs measure on the infinite group, then that consistency will tend to persist for a long time. We then return to the entropy-maximization problem raised in Chapter 2. We show that a maximal-entropy joining of two Gibbs measures for nearest-neighbor interactions (not necessarily the same interaction) must be a relative product over the tail σ-algebra, unless every joining has entropy minus infinity. In particular, if either is tail-trivial then the unique maximal-entropy joining is the product. In the latter case, this provides examples where the sofic entropy over a stochastic block model is equal to the f-invariant. We conclude by using recent results on bisections of random regular graphs to show that, for the free-boundary Ising model, the product self-joining has less than maximal f-invariant for some nontrivial temperature range.

ISBN: 9798516064456Subjects--Topical Terms:

3173530
Theoretical mathematics.
Subjects--Index Terms:

Ergodic theory

Entropy of Nonamenable Group Actions.
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100 1 $a Shriver, Christopher Edgar. $3 3686081
245 1 0 $a Entropy of Nonamenable Group Actions.
260 1 $a Ann Arbor : $b ProQuest Dissertations & Theses, $c 2021
300 $a 148 p.
500 $a Source: Dissertations Abstracts International, Volume: 82-12, Section: B.
500 $a Advisor: Austin, Timothy.
502 $a Thesis (Ph.D.)--University of California, Los Angeles, 2021.
506 $a This item must not be sold to any third party vendors.
520 $a Sofic entropy is an isomorphism invariant of measure-preserving actions of sofic groups introduced by Lewis Bowen around 2010. Its classical analogue was introduced in the 1950s by Kolmogorov and Sinai in order to show that Bernoulli shifts over ℤ are nonisomorphic when their base measures have different Shannon entropies. This entropy rate was actively studied over the next few decades and extended to arbitrary amenable groups by Ornstein and Weiss.On the one hand, amenable groups provide an appropriate setting for entropy theory since they have a way of performing the kind of average used to define an entropy rate. On the other hand, statistical physicists have long been interested in some nonamenable structures, such as the Bethe lattice. The problem of finding an appropriate entropy notion for nonamenable group actions, and in particular the problem of isomorphism of Bernoulli shifts in this setting, remained open until Bowen's work. One way to briefly summarize the idea of sofic entropy is to say that we consider the entropy per site along a sequence of large finite systems which locally approximate the infinite one (called a sofic approximation) rather than large finite subsystems of the infinite one. An interesting problem which arises is what effect the choice of sofic approximation can have on the sofic entropy rate.This thesis presents related work on several different problems of sofic entropy theory. In Chapter 2 we study the f-invariant, a variant of sofic entropy for free-group actions introduced by Bowen which can be defined using a kind of uniform random sofic approximation. We use a relative version of the f-invariant to show that the sofic entropy over a kind of "stochastic block model'' random sofic approximation is given by the solution to an entropy-maximization problem. Understanding this optimization problem may shed further light on the dependence of sofic entropy on the sofic approximation.Chapter 4 uses a new notion of sofic free energy density to study Gibbs measures and Glauber dynamics for nearest-neighbor interacting particle systems on some nonamenable groups. The main results are that, under certain reasonable conditions, every Glauber-invariant, shift-invariant measure is Gibbs and that the Glauber evolution of any shift-invariant measure converges to the set of Gibbs measures. These extend results of Holley (1971) for the Ising model on integer lattices.Chapter 5 begins by proving a metastability result for states on finite graphs which are locally similar to the Cayley graph of a finitely-generated group: the Glauber evolution of any state on a finite graph will converge to the unique Gibbs state, but we show that if the initial state is "pseudo-Gibbs'' in that it is in some sense consistent with some Gibbs measure on the infinite group, then that consistency will tend to persist for a long time. We then return to the entropy-maximization problem raised in Chapter 2. We show that a maximal-entropy joining of two Gibbs measures for nearest-neighbor interactions (not necessarily the same interaction) must be a relative product over the tail σ-algebra, unless every joining has entropy minus infinity. In particular, if either is tail-trivial then the unique maximal-entropy joining is the product. In the latter case, this provides examples where the sofic entropy over a stochastic block model is equal to the f-invariant. We conclude by using recent results on bisections of random regular graphs to show that, for the free-boundary Ising model, the product self-joining has less than maximal f-invariant for some nontrivial temperature range.
590 $a School code: 0031.
650 4 $a Theoretical mathematics. $3 3173530
650 4 $a Mathematics. $3 515831
653 $a Ergodic theory
653 $a Sofic groups
653 $a Sofic entropy theory
653 $a f-invariant
653 $a Free group actions
653 $a Free-energy density
653 $a Gibbs measures
653 $a Glauber dynamics
653 $a Metastability
653 $a Finite graphs
690 $a 0642
690 $a 0405
710 2 $a University of California, Los Angeles. $b Mathematics 0540. $3 2096468
773 0 $t Dissertations Abstracts International $g 82-12B.
790 $a 0031
791 $a Ph.D.
792 $a 2021
793 $a English
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28543597