arXiv subjects

Tim Austin

Publications and source records attributed to Tim Austin.

At least 19 recordsLinked to original sources

arXiv2025

Annealed almost periodic entropy

This work studies certain notions of entropy that can be associated to (i) a representation of a separable, unital C*-algebra $\mathfrak{A}$ and (ii) an auxiliary random sequence $(\pi_n)_{n\ge 1}$ of finite-dimensional representations of $\mathfrak{A}$. This continues a previous research program into the properties of these entropy notions when each $\pi_n$ is deterministic, which uncovered a range of analogies with entropy in ergodic theory and also with non-commutative generalizations of Szeg\H{o}'s limit theorems. We associate two new notions of entropy to data as in (i) and (ii) above: `annealed' AP entropy, which is roughly a kind of first-moment average of deterministic AP entropies; and `zeroth-order' AP entropy, which controls the large deviations probabilities that certain positive definite functions appear in the representations $\pi_n$ at all. After developing some of this general theory, we then focus on the special case in which $\mathfrak{A}$ is the group C*-algebra of a finitely-generated free group and each $\pi_n$ is generated by choosing a tuple of $n$-by-$n$ unitary matrices independently at random from Haar measure. In that case, explicit formulas can be derived for some of our notions of entropy, and new large deviations principles in random matrix theory are obtained as a consequence.

math.PR↗

arXiv2024

Entropy and determinants for unitary representations

Ergodic theory includes several notions of entropy for probability-preserving actions of countable groups. These include Kolmogorov--Sinai entropy based on F\o lner sequences for amenable groups, entropy defined using a random ordering of the group, and Bowen's sofic entropy for sofic groups. In this work we pursue these notions across an analogy between ergodic theory and representation theory. We arrive at new quantities associated to unitary representations of groups and representations of other C*-algebras. Our main results show that these new quantities can often be evaluated as Fuglede--Kadison determinants. The resulting determinantal formulas offer various non-commutative generalizations of Szeg\H{o}'s limit theorem for Toeplitz determinants. They make contact with Arveson's theory of subdiagonal subalgebras, and also with some entropy formulas in the ergodic theory of actions by automorphisms of compact Abelian groups.

math.OA↗

arXiv2024

Non-convergence of some non-commuting double ergodic averages

Let $S$ and $T$ be measure-preserving transformations of a probability space $(X,{\mathcal B},\mu)$. Let $f$ be a bounded measurable functions, and consider the integrals of the corresponding `double' ergodic averages: \[\frac{1}{n}\sum_{i=0}^{n-1} \int f(S^ix)f(T^ix)\ d\mu(x) \qquad (n\ge 1).\] We construct examples for which these integrals do not converge as $n\to\infty$. These include examples in which $S$ and $T$ are rigid, and hence have entropy zero, answering a question of Frantzikinakis and Host. Our proof begins with a corresponding construction for orthogonal operators on a Hilbert space, and then obtains transformations of a Gaussian measure space from them.

math.DS↗

arXiv2023

Algebraic dynamical systems from LDPC codes satisfy a strong negation of the weak Pinsker property

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 a direct Bernoulli factor, because its model spaces shatter into exponentially many clusters of sub-exponential size. The example and its analysis are related to random low-density parity-check (LDPC) codes.

math.DS↗

arXiv2021

An ergodic system is dominant exactly when it has positive entropy

An ergodic dynamical system $\mathbf{X}$ is called dominant if it is isomorphic to a generic extension of itself. It was shown in an earlier paper by Glasner, Thouvenot and Weiss that Bernoulli systems with finite entropy are dominant. In this work we show first that every ergodic system with positive entropy is dominant, and then that if $\mathbf{X}$ has zero entropy then it is not dominant.

math.DS↗

arXiv2020

A new dynamical proof of the Shmerkin--Wu theorem

Let $a < b$ be multiplicatively independent integers, both at least $2$. Let $A,B$ be closed subsets of $[0,1]$ that are forward invariant under multiplication by $a$, $b$ respectively, and let $C := A\times B$. An old conjecture of Furstenberg asserted that any planar line $L$ not parallel to either axis must intersect $C$ in Hausdorff dimension at most $\max\{\dim C,1\} - 1$. Two recent works by Shmerkin and Wu have given two different proofs of this conjecture. This note provides a third proof. Like Wu's, it stays close to the ergodic theoretic machinery that Furstenberg introduced to study such questions, but it uses less substantial background from ergodic theory. The same method is also used to re-prove a recent result of Yu about certain sequences of sums.

math.DS↗

arXiv2018

The structure of low-complexity Gibbs measures on product spaces

Let $K_1$, $\dots$, $K_n$ be bounded, complete, separable metric spaces. Let $\lambda_i$ be a Borel probability measure on $K_i$ for each $i$. Let $f:\prod_i K_i \to \mathbb{R}$ be a bounded and continuous potential function, and let $$\mu(d \mathbf{x})\ \propto\ e^{f(\mathbf{x})}\lambda_1(d x_1)\cdots \lambda_n(d x_n)$$ be the associated Gibbs distribution. At each point $\mathbf{x} \in \prod_i K_i$, one can define a `discrete gradient' $\nabla f(\mathbf{x},\,\cdot\,)$ by comparing the values of $f$ at all points which differ from $\mathbf{x}$ in at most one coordinate. In case $\prod_i K_i = \{-1,1\}^n \subset \mathbb{R}^n$, the discrete gradient $\nabla f(\mathbf{x},\,\cdot\,)$ is naturally identified with a vector in $\mathbb{R}^n$. This paper shows that a `low-complexity' assumption on $\nabla f$ implies that $\mu$ can be approximated by a mixture of other measures, relatively few in number, and most of them close to product measures in the sense of optimal transport. This implies also an approximation to the partition function of $f$ in terms of product measures, along the lines of Chatterjee and Dembo's theory of `nonlinear large deviations'. An important precedent for this work is a result of Eldan in the case $\prod_i K_i = \{-1,1\}^n$. Eldan's assumption is that the discrete gradients $\nabla f(\mathbf{x},\,\cdot\,)$ all lie in a subset of $\mathbb{R}^n$ that has small Gaussian width. His proof is based on the careful construction of a diffusion in $\mathbb{R}^n$ which starts at the origin and ends with the desired distribution on the subset $\{-1,1\}^n$. Here our assumption is a more naive covering-number bound on the set of gradients $\{\nabla f(\mathbf{x},\,\cdot\,):\ \mathbf{x} \in \prod_i K_i\}$, and our proof relies only on basic inequalities of information theory. As a result, it is shorter, and applies to Gibbs measures on arbitrary product spaces.

math.PR↗

arXiv2018

Multi-variate correlation and mixtures of product measures

Total correlation (`TC') and dual total correlation (`DTC') are two classical ways to quantify the correlation among an $n$-tuple of random variables. They both reduce to mutual information when $n=2$. The first part of this paper sets up the theory of TC and DTC for general random variables, not necessarily finite-valued. This generality has not been exposed in the literature before. The second part considers the structural implications when a joint distribution $\mu$ has small TC or DTC. If $\mathrm{TC}(\mu) = o(n)$, then $\mu$ is close to a product measure according to a suitable transportation metric: this follows directly from Marton's classical transportation-entropy inequality. If $\mathrm{DTC}(\mu) = o(n)$, then the structural consequence is more complicated: $\mu$ is a mixture of a controlled number of terms, most of them close to product measures in the transportation metric. This is the main new result of the paper.

math.PR↗

arXiv2017

Gibbs measures over locally tree-like graphs and percolative entropy over infinite regular trees

Consider a statistical physical model on the $d$-regular infinite tree $T_{d}$ described by a set of interactions $\Phi$. Let $\{G_{n}\}$ be a sequence of finite graphs with vertex sets $V_n$ that locally converge to $T_{d}$. From $\Phi$ one can construct a sequence of corresponding models on the graphs $G_n$. Let $\{\mu_n\}$ be the resulting Gibbs measures. Here we assume that $\{\mu_{n}\}$ converges to some limiting Gibbs measure $\mu$ on $T_{d}$ in the local weak$^*$ sense, and study the consequences of this convergence for the specific entropies $|V_n|^{-1}H(\mu_n)$. We show that the limit supremum of $|V_n|^{-1}H(\mu_n)$ is bounded above by the \emph{percolative entropy} $H_{perc}(\mu)$, a function of $\mu$ itself, and that $|V_n|^{-1}H(\mu_n)$ actually converges to $H_{perc}(\mu)$ in case $\Phi$ exhibits strong spatial mixing on $T_d$. We discuss a few examples of well-known models for which the latter result holds in the high temperature regime.

math.PR↗

arXiv2017

Measure concentration and the weak Pinsker property

Let $(X,\mu)$ be a standard probability space. An automorphism $T$ of $(X,\mu)$ has the weak Pinsker property if for every $\varepsilon > 0$ it has a splitting into a direct product of a Bernoulli shift and an automorphism of entropy less than $\varepsilon$. This property was introduced by Thouvenot, who asked whether it holds for all ergodic automorphisms. This paper proves that it does. The proof actually gives a more general result. Firstly, it gives a relative version: any factor map from one ergodic automorphism to another can be enlarged by arbitrarily little entropy to become relatively Bernoulli. Secondly, using some facts about relative orbit equivalence, the analogous result holds for all free and ergodic measure-preserving actions of a countable amenable group. The key to this work is a new result about measure concentration. Suppose now that $\mu$ is a probability measure on a finite product space $A^n$, and endow this space with its Hamming metric. We prove that $\mu$ may be represented as a mixture of other measures in which (i) most of the weight in the mixture is on measures that exhibit a strong kind of concentration, and (ii) the number of summands is bounded in terms of the difference between the Shannon entropy of $\mu$ and the combined Shannon entropies of its marginals.

math.DS↗

arXiv2017

An asymptotic equipartition property for measures on model spaces

Let $G$ be a sofic group, and let $\Sigma = (\sigma_n)_{n\geq 1}$ be a sofic approximation to it. For a probability-preserving $G$-system, a variant of the sofic entropy relative to $\Sigma$ has recently been defined in terms of sequences of measures on its model spaces that `converge' to the system in a certain sense. Here we prove that, in order to study this notion, one may restrict attention to those sequences that have the asymptotic equipartition property. This may be seen as a relative of the Shannon--McMillan theorem in the sofic setting. We also give some first applications of this result, including a new formula for the sofic entropy of a $(G\times H)$-system obtained by co-induction from a $G$-system, where $H$ is any other infinite sofic group.

math.DS↗

arXiv2016

Behaviour of entropy under bounded and integrable orbit equivalence

Let $G$ and $H$ be infinite finitely generated amenable groups. This paper studies two notions of equivalence between actions of such groups on standard Borel probability spaces. They are defined as stable orbit equivalences in which the associated cocycles satisfy certain tail bounds. In `integrable stable orbit equivalence', the length in $H$ of the cocycle-image of an element of $G$ must have finite integral over its domain (a subset of the $G$-system), and similarly for the reverse cocycle. In `bounded stable orbit equivalence', these functions must be essentially bounded in terms of the length in $G$. `Integrable' stable orbit equivalence arises naturally in the study of integrable measure equivalence of groups themselves, as introduced recently by Bader, Furman and Sauer. The main result is a formula relating the Kolmogorov--Sinai entropies of two actions which are equivalent in one of these ways. Under either of these tail assumptions, the entropies stand in a proportion given by the compression constant of the stable orbit equivalence. In particular, in the case of full orbit equivalence subject to such a tail bound, entropy is an invariant. This contrasts with the case of unrestricted orbit equivalence, under which all free ergodic actions of countable amenable groups are equivalent. The proof uses an entropy-bound based on graphings for orbit equivalence relations, and in particular on a new notion of cost which is weighted by the word lengths of group elements.

math.DS↗

arXiv2016

Uniform mixing and completely positive sofic entropy

Let $G$ be a countable discrete sofic group. We define a concept of uniform mixing for measure-preserving $G$-actions and show that it implies completely positive sofic entropy. When $G$ contains an element of infinite order, we use this to produce an uncountable family of pairwise nonisomorphic $G$-actions with completely positive sofic entropy. None of our examples is a factor of a Bernoulli shift.

math.DS↗

arXiv2015

The geometry of model spaces for probability-preserving actions of sofic groups

Bowen's notion of sofic entropy is a powerful invariant for classifying probability-preserving actions of sofic groups. It can be defined in terms of the covering numbers of certain metric spaces associated to such an action, the `model spaces'. The metric geometry of these model spaces can exhibit various interesting features, some of which provide other invariants of the action. This paper explores an approximate connectedness property of the model spaces, and uses it give a new proof that certain groups admit factors of Bernoulli shifts which are not Bernoulli. This was originally proved by Popa. Our proof covers fewer examples than his, but provides additional information about this phenomenon.

math.DS↗

arXiv2015

Additivity properties of sofic entropy and measures on model spaces

Sofic entropy is an invariant for probability-preserving actions of sofic groups. It was introduced a few years ago by Lewis Bowen, and shown to extend the classical Kolmogorov-Sinai entropy from the setting of amenable groups. Some parts of Kolmogorov-Sinai entropy theory generalize to sofic entropy, but in other respects this new invariant behaves less regularly. This paper explores conditions under which sofic entropy is additive for Cartesian products of systems. It is always subadditive, but the reverse inequality can fail. We define a new entropy-notion in terms of probability distributions on the spaces of good models of an action. Using this, we prove a general lower bound for the sofic entropy of a Cartesian product in terms of separate quantities for the two factor systems involved. We also prove that this lower bound is optimal in a certain sense, and use it to derive some sufficient conditions for the strict additivity of sofic entropy itself. Various other properties of this new entropy notion are also developed.

math.DS↗

arXiv2015

Ajtai-Szemer\'edi Theorems over quasirandom groups

Two versions of the Ajtai-Szemer\'edi Theorem are considered in the Cartesian square of a finite non-Abelian group $G$. In case $G$ is sufficiently quasirandom, we obtain strong forms of both versions: if $E \subseteq G\times G$ is fairly dense, then $E$ contains a large number of the desired patterns for most individual choices of `common difference'. For one of the versions, we also show that this set of good common differences is syndetic.

math.CO↗

arXiv2014

Pleasant extensions retaining algebraic structure, II

In this paper we combine the general tools developed in (arXiv:0905.0518) with several ideas taken from earlier work on one-dimensional nonconventional ergodic averages by Furstenberg and Weiss, Host and Kra and Ziegler to study the averages $\frac{1}{N}\sum_{n=1}^N(f_1\circ T^{n\bf{p}_1})(f_2\circ T^{n\bf{p}_2})(f_3\circ T^{n\bf{p}_3})$ for $f_1,f_2,f_3 \in L^\infty(μ)$ associated to a triple of directions $\bf{p}_1,\bf{p}_2,\bf{p}_3 \in \mathbb{Z}^2$ that lie in general position along with $0 \in \mathbb{Z}^2$. We will show how to construct a `pleasant' extension of an initially-given $\mathbb{Z}^2$-system for which these averages admit characteristic factors with a very concrete description, involving one-dimensional isotropy factors and two-step pro-nilsystems. We also use this analysis to construct pleasant extensions and then prove norm convergence for the polynomial nonconventional ergodic averages $\frac{1}{N}\sum_{n=1}^N(f_1\circ T_1^{n^2})(f_2\circ T_1^{n^2}T_2^n)$ associated to two commuting transformations $T_1$, $T_2$.

math.DS↗

arXiv2014

Pleasant extensions retaining algebraic structure, I

In two recent papers we introduced some new techniques for constructing an extension of a probability-preserving system $T:\mathbb{Z}^d\curvearrowright (X,μ)$ that enjoys certain desirable properties in connexion with the asymptotic behaviour of some related nonconventional ergodic averages. The present paper is the first of two that will explore various refinements and extensions of these ideas. This first part is dedicated to some much more general machinery for the construction of extensions that can be used to recover various earlier results. It also contains two relatively simple new applications of this machinery to the study of certain families of nonconventional averages, one in discrete and one in continuous time (convergence being a new result for the latter). In the forthcoming second part (arXiv:0910.0907) we will introduce the problem of describing the characteristic factors and the limit of the linear nonconventional averages $\frac{1}{N}\sum_{n=1}^N \prod_{i=1}^kf_i\circ T^{n\bf{p}_i}$ when the directions $\bf{p}_1$, $\bf{p}_2$, \ldots, $\bf{p}_k \in \mathbb{Z}^d$ are not assumed to be linearly independent, and provide a fairly detailed solution in the case when k = 3, d = 2 and any pair of directions is linearly independent. This will then be used to prove the convergence in $L^2(μ)$ of the quadratic nonconventional averages $\frac{1}{N}\sum_{n=1}^N (f_1\circ T_1^{n^2})(f_2\circ T_1^{n^2}T_2^n)$.

math.DS↗