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Ádám Timár

Publications and source records attributed to Ádám Timár.

At least 19 recordsLinked to original sources
arXiv2026

The Almost Finite--Purely Infinite Dichotomy for Minimal Amenable Ample Groupoids

We answer, in the principal setting, a question of Matui on the almost finite--purely infinite dichotomy for minimal amenable ample groupoids. We prove that a second-countable Hausdorff minimal principal topologically amenable ample groupoid with Cantor unit space is strongly almost finite if and only if it admits an invariant probability measure; if no such measure exists, then it is purely infinite. We also prove that every bounded-degree uniformly Borel amenable Følner graph is Borel almost finite. The proofs combine the recent three-to-two comparison method of Glasner and Liu with the randomized Følner packing methods of Elek and Timár.

math.DS↗
arXiv2026

Uniform Borel Amenability

We study a uniform, quantitative form of the amenability-hyperfiniteness paradigm for bounded-degree Borel graphs generating countable Borel equivalence relations. We introduce \emph{uniform Borel amenability} and prove that it is equivalent to \emph{randomized Borel hyperfiniteness}, a probabilistic version of hyperfiniteness. Consequences are three strengthenings of the Connes-Feldman-Weiss theorem. In the setting of uniformly Borel amenable Følner graphs (e.g. Borel graphs of not necessarily free actions of amenable groups or Borel graphs of subexponential growth), we establish an analogous equivalence to randomized Borel almost finiteness. We further obtain measure-theoretic structural results, including almost finiteness outside a $μ$-null invariant set extending a recent result of Conley et al. for free amenable actions, and an Ornstein--Weiss type packing theorem that is uniform over all invariant measures. Finally, we show that uniformly Borel amenable graphs are hyperfinite modulo a compressible invariant set, i.e., after removing a Borel invariant set that is of measure zero for every invariant probability measure.

math.DS↗
arXiv2026

Parabolic-hyperbolic dichotomy through half-plane coexistence

Consider a unimodular random planar map (URM) with an invariant ergodic percolation having infinite primal and dual clusters. We say that there is half-plane coexistence if both the percolation and its dual have infinite clusters when restricted to a half-plane. Under mild assumptions on the percolation, we show that the URM is parabolic if and only if there is no half-plane coexistence, and it is hyperbolic if and only if there is half-plane coexistence. This extends the recent half-plane non-coexistence result for $\mathbb{Z}^2$ by Klausen and Kravitz and provides another manifestation of the parabolic-hyperbolic dichotomy for URM's.

math.PR↗
arXiv2025

Indistinguishability for recurrent clusters

We introduce a general framework to show the indistinguishability of infinite clusters (ergodicity of the cluster subrelation) in group-invariant percolation processes with a weaker version of the finite energy property: the possibility of moving infinite branches from one infinite cluster to another. Crucially, this removes the necessity for the infinite clusters to be transient, present in most previous works. Our method also applies to more general random graphs, whenever a stationary sequence of vertices is definable. We use this to show the indistinguishability of infinite clusters (or permutation cycles) in the interchange process (a.k.a.~random stirring process), the loop $O(n)$ model on amenable Cayley graphs, biased corner percolation on $\mathbb{Z}^2$, and the Poisson Zoo process. Finally, we show that infinite clusters in any invariant process on a Cayley graph are indistinguishable for any ``not essentially tail'' property, i.e., properties that depend only on the local structure of the cluster.

math.PR↗
arXiv2025

Factor of iid's through stochastic domination

We develop a method to prove that certain percolation processes on amenable random rooted graphs are factors of iid (fiid), given that the process is a monotone limit of random finite subgraphs that satisfy a certain independent stochastic domination property. Among the consequences are the previously open claims that the Uniform Spanning Forest (USF) is a factor of iid for recurrent graphs, it is a finite-valued finitary fiid on amenable graphs, and that the critical Ising model on $\Z^d$ is a finite-valued finitary fiid, using the known uniqueness of the Gibbs measure.

math.PR↗
arXiv2025

Strong almost finiteness

A countable, bounded degree graph is almost finite if it has a tiling with isomorphic copies of finitely many Følner sets, and we call it strongly almost finite, if the tiling can be randomized so that the probability that a vertex is on the boundary of a tile is uniformly small. We give various equivalents for strong almost finiteness. In particular, we prove that Property A together with the Følner property is equivalent to strong almost finiteness. Using these characterizations, we show that graphs of subexponential growth and Schreier graphs of amenable groups are always strongly almost finite, generalizing the celebrated result of Downarowicz, Huczek and Zhang about amenable Cayley graphs, based on graph theoretic rather than group theoretic principles. We give various equivalents to Property A for graphs, and show that if a sequence of graphs of Property A (in a uniform sense) converges to a graph $G$ in the neighborhood distance (a purely combinatorial analogue of the classical Benjamini-Schramm distance), then their Laplacian spectra converge to the Laplacian spectrum of $G$ in the Hausdorff distance. We apply the previous theory to construct a new and rich class of classifiable $C^{\star}$-algebras. Namely, we show that for any minimal strong almost finite graph $G$ there are naturally associated simple, nuclear, stably finite $C^{\star}$-algebras that are classifiable by their Elliott invariants.

math.GR↗
arXiv2025

Weighted-amenability and percolation

In 1999, Benjamini, Lyons, Peres, and Schramm introduced a notion of weighted-amenability for transitive graphs that is equivalent to the amenability of its automorphism group. For unimodular graphs this notion coincides with classical graph-amenability and has been intensely studied. In the present work, we show that many classical unimodular results can be extended to the nonunimodular setting, which is further motivated by recent progress in the mcp (measure class preserving or quasi-pmp) setting of measured group theory. To this end, we prove new characterizations of weighted-amenability, in particular that it is equivalent to all finite unions of levels inducing amenable graphs. Hutchcroft conjectured that the latter property implies that $p_h<p_u$, where $p_h$ is the critical probability for the regime where clusters of Bernoulli percolation are infinite total weight and $p_u$ is the uniqueness threshold. We prove a relaxed version of his conjecture à la Pak--Smirnova-Nagnibeda. Further characterizations are given in terms of the spectral radius and invariant spanning forests. One of the consequences is the continuity of the phase transition at $p_h$ for weighted-nonamenable graphs.

math.PR↗
arXiv2024

A unimodular random graph with large upper growth and no growth

We construct a unimodular random rooted graph with maximal degree $d\geq 3$ and upper growth rate $d-1$, which does not have a growth rate. Abért, Fraczyk and Hayes showed that for a unimodular random tree, if the upper growth rate is at least $\sqrt{d-1}$, then the growth rate exists, and asked with some scepticism if this may hold for more general graphs. Our construction shows that the answer is negative. We also provide a non-hyperfinite example of a unimodular random graph with no growth rate. This may be of interest in light of a conjecture of Abért that unimodular Riemannian surfaces of bounded negative curvature always have growth.

math.PR↗
arXiv2024

Physical networks as network-of-networks

Physical networks are made of nodes and links that are physical objects embedded in a geometric space. Understanding how the mutual volume exclusion between these elements affects the structure and function of physical networks calls for a suitable generalization of network theory. Here, we introduce a network-of-networks framework where we describe the shape of each extended physical node as a network embedded in space and these networks are bound together by physical links. Relying on this representation, we introduce a minimal model of network growth and we show for a general class of physical networks that volume exclusion induces heterogeneity in both node volume and degree, with the two becoming correlated. These emergent properties strongly affect the dynamics on physical networks: by calculating their Laplacian spectrum as a function of the coupling strength between the nodes we show that degree-volume correlations suppress the role of hubs as early spreaders in diffusive dynamics. We apply the network-of-networks framework to describe several real systems and find properties analog to the minimal model networks. The prevalence of these properties points towards general growth mechanisms that do not depend on the specifics of the systems.

cond-mat.stat-mech↗
arXiv2024

Finitely dependent random colorings of bounded degree graphs

We prove that every (possibly infinite) graph of degree at most $d$ has a 4-dependent random proper $4^{d(d+1)/2}$-coloring, and one can construct it as a finitary factor of iid. For unimodular transitive (or unimodular random) graphs we construct an automorphism-invariant (respectively, unimodular) 2-dependent coloring by $3^{d(d+1)/2}$ colors. In particular, there exist random proper colorings for $\Z^d$ and for the regular tree that are 2-dependent and automorphism-invariant, or 4-dependent and finitary factor of iid.

math.PR↗
arXiv2023

A full characterization of invariant embeddability of unimodular planar graphs

When can a unimodular random planar graph be drawn in the Euclidean or the hyperbolic plane in a way that the distribution of the random drawing is isometry-invariant? This question was answered for one-ended unimodular graphs in \cite{benjamini2019invariant}, using the fact that such graphs automatically have locally finite (simply connected) drawings into the plane. For the case of graphs with multiple ends the question was left open. We revisit Halin's graph theoretic characterization of graphs that have a locally finite embedding into the plane. Then we prove that such unimodular random graphs do have a locally finite invariant embedding into the Euclidean or the hyperbolic plane, depending on whether the graph is amenable or not.

math.PR↗
arXiv2023

On invariant generating sets for the cycle space

Consider a unimodular random graph, or just a finitely generated Cayley graph. When does its cycle space have an invariant random generating set of cycles such that every edge is contained in finitely many of the cycles? Generating the free Loop $O(1)$ model as a factor of iid is closely connected to having such a generating set for FK-Ising cluster. We show that geodesic cycles do not always form such a generating set, by showing for a parameter regime of the FK-Ising model on the lamplighter group the origin is contained in infinitely many geodesic cycles. This answers a question by Angel, Ray and Spinka. Then we take a look at how the property of having an invariant locally finite generating set for the cycle space is preserved by Bernoulli percolation, and apply it to the problem of factor of iid presentations of the free Loop $O(1)$ model.

math.PR↗
arXiv2021

The Free Uniform Spanning Forest is disconnected in some virtually free groups, depending on the generator set

We prove the rather counterintuitive result that there exist finite transitive graphs H and integers k such that the Free Uniform Spanning Forest in the direct product of the k-regular tree and H has infinitely many trees almost surely. This shows that the number of trees in the FUSF is not a quasi-isometry invariant. Moreover, we give two different Cayley graphs of the same virtually free group such that the FUSF has infinitely many trees in one, but is connected in the other, answering a question of Lyons and Peres (2016) in the negative. A version of our argument gives an example of a non-unimodular transitive graph where WUSF\not=FUSF, but some of the FUSF trees are light with respect to Haar measure. This disproves a conjecture of Tang (2019).

math.PR↗
arXiv2021

Finite-energy infinite clusters without anchored expansion

Hermon and Hutchcroft have recently proved the long-standing conjecture that in Bernoulli(p) bond percolation on any nonamenable transitive graph G, at any p > p_c(G), the probability that the cluster of the origin is finite but has a large volume n decays exponentially in n. A corollary is that all infinite clusters have anchored expansion almost surely. They have asked if these results could hold more generally, for any finite energy ergodic invariant percolation. We give a counterexample, an invariant percolation on the 4-regular tree.

math.PR↗
arXiv2020

Connectedness of the Free Uniform Spanning Forest as a function of edge weights

Let $G$ be the Cartesian product of a regular tree $T$ and a finite connected transitive graph $H$. It is shown in arXiv:2006.06387 that the Free Uniform Spanning Forest ($\mathsf{FSF}$) of this graph may not be connected, but the dependence of this connectedness on $H$ remains somewhat mysterious. We study the case when a positive weight $w$ is put on the edges of the $H$-copies in $G$, and conjecture that the connectedness of the $\mathsf{FSF}$ exhibits a phase transition. For large enough $w$ we show that the $\mathsf{FSF}$ is connected, while for a large family of $H$ and $T$, the $\mathsf{FSF}$ is disconnected when $w$ is small (relying on arXiv:2006.06387). Finally, we prove that when $H$ is the graph of one edge, then for any $w$, the $\mathsf{FSF}$ is a single tree, and we give an explicit formula for the distribution of the distance between two points within the tree.

math.PR↗
arXiv2018

Controllability, matching ratio and graph convergence

There is an important parameter in control theory which is closely related to the directed matching ratio of the network, as shown by Liu, Slotine and Barabási (2011). We give proofs on two main statements of that paper on the directed matching ratio, which were based on numerical results and heuristics from statistical physics. First, we show that the directed matching ratio of directed random networks given by a fix sequence of degrees is concentrated around its mean. We also examine the convergence of the (directed) matching ratio of a random (directed) graph sequence that converges in the local weak sense, and generalize the result of Elek and Lippner (2009). We prove that the mean of the directed matching ratio converges to the properly defined matching ratio parameter of the limiting graph. We further show the almost sure convergence of the matching ratios for the most widely used families of scale-free networks, which was the main motivation of Liu, Slotine and Barabási.

math.PR↗
arXiv2017

On percolation critical probabilities and unimodular random graphs

We investigate generalisations of the classical percolation critical probabilities $p_c$, $p_T$ and the critical probability $\tilde{p_c}$ defined by Duminil-Copin and Tassion (2015) to bounded degree unimodular random graphs. We further examine Schramm's conjecture in the case of unimodular random graphs: does $p_c(G_n)$ converge to $p_c(G)$ if $G_n\to G$ in the local weak sense? Among our results are the following: 1. $p_c=\tilde{p_c}$ holds for bounded degree unimodular graphs. However, there are unimodular graphs with sub-exponential volume growth and $p_T < p_c$; i.e., the classical sharpness of phase transition does not hold. 2. We give conditions which imply $\lim p_c(G_n) = p_c(\lim G_n)$. 3. There are sequences of unimodular graphs such that $G_n\to G$ but $p_c(G)>\lim p_c(G_n)$ or $p_c(G)<\lim p_c(G_n)<1$. As a corollary to our positive results, we show that for any transitive graph with sub-exponential volume growth there is a sequence $T_n$ of large girth bi-Lipschitz invariant subgraphs such that $p_c(T_n)\to 1$. It remains open whether this holds whenever the transitive graph has cost 1.

math.PR↗