arXiv subjects
Publications and source records attributed to Lewis Bowen.
The hard sphere model is a classical model from statistical physics in which particles are represented by equal-sized spheres. Longstanding predictions from the physics literature indicate that in $\mathbb{R}^2$ and $\mathbb{R}^3$ the system undergoes a phase transition, but it remains a major open problem to confirm this. We prove the existence of a phase transition for this model in the hyperbolic plane.
A group is surjunctive if every injective cellular automaton on it is also surjective. Gottschalk famously conjectured that all groups are surjunctive. This remains a central open problem in symbolic dynamics and descriptive set theory. Gromov and Weiss termed the notion of sofic groups, and proved that all such groups are surjunctive, providing the largest class of groups which satisfy Gottschalk's conjecture. It is still open to decide whether all groups are sofic. This became a major open problem in group theory, and is related to other well known problems such as the Aldous--Lyons conjecture in probability theory and to Connes' embedding problem in the theory of operator algebras. A complementary natural question to ask is: Does the reverse implication to Gromov and Weiss' result holds? Namely, are all surjunctive groups sofic? As currently there are no known non-sofic groups, answering this problem in the negative in the category of groups is still out of reach. This paper resolves this problem in the generalized setup of invariant random subgroups of free groups (IRSs), where non (co)sofic objects were recently shown to exist by Lubotzky, Vidick and the two authors. Specifically, we prove that there exists a surjunctive non (co)sofic IRS, resolving the aforementioned problem in the negative. Our proof uses a complexity theoretic approach, and in particular a recent development due to Manzoor, as well as the theory of Rokhlin entropy developed by Seward and others. As a byproduct of our proof technique, the non (co)sofic IRS we provide satisfies a condition stronger than surjunctivity; it satisfies a version of Seward's maximal Rokhlin entropy of Bernoulli Shifts (RBS) criterion.
We establish general criteria for a countable group $\Gamma$ to have fixed price 1 depending on a choice of left-invariant proper metric on $\Gamma$. We apply this criterion to show that if $\Gamma_1,\Gamma_2$ are two countable groups satisfying a certain growth condition then $\Gamma_1\times \Gamma_2$ has fixed price 1. For example, $\Gamma\times \Gamma$ has fixed price 1 for any countable group $\Gamma$.
We prove that the sequence of Masur-Smillie-Veech (MSV) distributed random translation surfaces, with area equal to genus, Benjamini-Schramm converges as genus tends to infinity. This means that for any fixed radius $r>0$, if $X_g$ is an MSV-distributed random translation surface with area $g$ and genus $g$, and $o$ is a uniformly random point in $X_g$, then the radius-$r$ neighborhood of $o$ in $X_g$, as a pointed measured metric space, converges in distribution to the radius $r$ neighborhood of the root in a Poisson translation plane, which is a random pointed surface we introduce here. Along the way, we obtain bounds on statistical local geometric properties of translation surfaces, such as the probability that the random point $o$ has injectivity radius at most $r$, which may be of independent interest.
This paper, and its companion [BCLV24], are devoted to a negative resolution of the Aldous--Lyons Conjecture [AL07, Ald07]. In this part we study tailored non-local games. This is a subclass of non-local games -- combinatorial objects which model certain experiments in quantum mechanics, as well as interactive proofs in complexity theory. Our main result is that, given a tailored non-local game $G$, it is undecidable to distinguish between the case where $G$ has a special kind of perfect strategy, and the case where every strategy for $G$ is far from being perfect. Using a reduction introduced in the companion paper [BCLV24], this undecidability result implies a negative answer to the Aldous--Lyons conjecture. Namely, it implies the existence of unimodular networks that are non-sofic. To prove our result, we use a variant of the compression technique developed in MIP*=RE [JNV+21]. Our main technical contribution is to adapt this technique to the class of tailored non-local games. The main difficulty is in establishing answer reduction, which requires a very careful adaptation of existing techniques in the construction of probabilistically checkable proofs. As a byproduct, we are reproving the negation of Connes' embedding problem [Con76] -- i.e., the existence of a $\mathrm{II}_1$-factor which cannot be embedded in an ultrapower of the hyperfinite $\mathrm{II}_1$-factor -- first proved in [JNV+21], using an arguably more streamlined proof. In particular, we incorporate recent simplifications from the literature [dlS22b, Vid22] due to de la Salle and the third author.
This paper, and its companion [BCV24], are devoted to a negative resolution of the Aldous--Lyons Conjecture [AL07, Ald07]. This conjecture, originated in probability theory, is well known (cf. [Gel18]) to be equivalent to the statement that every invariant random subgroup of the free group is co-sofic. We disprove this last statement. In this part we introduce subgroup tests. These tests are finite distributions over continuous functions from the space of subgroups of the free group to $\{0,1\}$. Subgroup tests provide a general framework in which one can study invariant random subgroups of the free group. Classical notions such as group soficity and group stability arise naturally in this framework. By the correspondence between subgroups of the free group and Schreier graphs, one can view subgroup tests as a property testing model for certain edge-labeled graphs. This correspondence also provides the connection to random networks. Subgroup tests have values, which are their asymptotic optimal expectations when integrated against co-sofic invariant random subgroups. Our first main result is that, if every invariant random subgroup of the free group is co-sofic, then one can approximate the value of a subgroup test up to any positive additive constant. Our second main result is an essentially value preserving correspondence between certain non-local games and subgroup tests. By composing this correspondence with a stronger variant of the reduction in MIP*=RE [JNV+21], proved in the companion paper [BCV24], we deduce that approximating the sofic value of a subgroup test is as hard as the Halting Problem, and in particular, undecidable. The combination of our two main results proves the existence of non co-sofic invariant random subgroups of the free group.
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.
This paper generalizes sofic entropy theory, in both the topological and measure-theory settings, to actions of locally compact groups. We prove invariance under topological and measure conjugacy of these entropies and establish the variational principle.
The $f$-invariant is a notion of entropy for probability-measure-preserving actions of free groups. We show it is invariant under bounded orbit-equivalence.
A sofic approximation to a countable group is a sequence of partial actions on finite sets that asymptotically approximates the action of the group on itself by left-translations. A group is sofic if it admits a sofic approximation. Sofic entropy theory is a generalization of classical entropy theory in dynamics to actions by sofic groups. However, the sofic entropy of an action may depend on a choice of sofic approximation. All previously known examples showing this dependence rely on degenerate behavior. This paper exhibits an explicit example of a mixing subshift of finite type with two different positive sofic entropies. The example is inspired by statistical physics literature on 2-colorings of random hyper-graphs.
We introduce the notion of soficity for locally compact groups and list a number of open problems.
We show that the class $\mathscr{B}$, of discrete groups which satisfy the conclusion of Popa's Cocycle Superrigidity Theorem for Bernoulli actions, is invariant under measure equivalence. We generalize this to the setting of discrete p.m.p. groupoids, and as a consequence we deduce that any nonamenable lattice in a product of two noncompact, locally compact second countable groups, must belong to $\mathscr{B}$. We also introduce a measure-conjugacy invariant called Weak Pinsker entropy and show that, if G is a group in the class $\mathscr{B}$, then Weak Pinsker entropy is an orbit-equivalence invariant of every essentially free p.m.p. action of G.
A probability-measure-preserving transformation has the Weak Pinsker Property (WPP) if for every $ε>0$ it is measurably conjugate to the direct product of a transformation with entropy $<ε$ and a Bernoulli shift. In a recent breakthrough, Tim Austin proved that every ergodic transformation satisfies this property. Moreover, the natural analog for amenable group actions is also true. By contrast, this paper provides a counterexample in which the group $Γ$ is a non-abelian free group and the notion of entropy is sofic entropy. The counterexample is a limit of hardcore models on random regular graphs. In order to prove that it does not have the WPP, this paper introduces new measure conjugacy invariants based on the growth of homology of the model spaces of the action. The main result is obtained by showing that any action with the WPP has subexponential homology growth in dimension $0$, while the counterexample has exponential homology growth in dimension $0$.
The classical Multiplicative Ergodic Theorem (MET) of Oseledets is generalized here to cocycles taking values in a semi-finite von Neumann algebra. This allows for a continuous Lyapunov distribution.
A sofic group $G$ is said to be flexibly stable if every sofic approximation to $G$ can converted to a sequence of disjoint unions of Schreier graphs by modifying an asymptotically vanishing proportion of edges. We establish that if $\mathrm{PSL}_d(\mathbb{Z})$ is flexibly stable for some $d \geq 5$ then there exists a group which is not sofic.
Amos Nevo established the pointwise ergodic theorem in $L^p$ for measure-preserving actions of $\mathrm{PSL}_2(\mathbb{R})$ on probability spaces with respect to ball averages and every $p>1$. This paper shows by explicit example that Nevo's Theorem cannot be extended to $p=1$.
The von Neumann-Day problem asks whether every non-amenable group contains a non-abelian free group. It was answered in the negative by Ol'shanskii in the 1980s. The measurable version (formulated by Gaboriau-Lyons) asks whether every non-amenable measured equivalence relation contains a non-amenable treeable subequivalence relation. This paper obtains a positive answer in the case of arbitrary Bernoulli shifts over a non-amenable group, extending work of Gaboriau-Lyons. The proof uses an approximation to the random interlacement process by random multistep of geometrically-killed random walk paths. There are two applications: (1) the Gaboriau-Lyons problem for actions with positive Rokhlin entropy admits a positive solution, (2) for any non-amenable group, all Bernoulli shifts factor onto each other.
Kolmogorov-Sinai entropy is an invariant of measure-preserving actions of the group of integers that is central to classification theory. There are two recently developed invariants, sofic entropy and Rokhlin entropy, that generalize classical entropy to actions of countable groups. These new theories have counterintuitive properties such as factor maps that increase entropy. This survey article focusses on examples, many of which have not appeared before, that highlight the differences and similarities with classical theory.