arXiv subjects

László Márton Tóth

Publications and source records attributed to László Márton Tóth.

13 recordsLinked to original sources

arXiv2026

Whitney's 2-isomorphism theorem for graphings

We prove measurable analogues of Whitney's classical theorems on weak isomorphisms of finite graphs. For locally finite graphings, we define a weak isomorphism to be an edge-measure-preserving Borel bijection that preserves cycles and hyperfinite subgraphs, modulo null sets. Our rigidity theorem shows that every weak isomorphism of a weakly $3$-connected, infinitely-ended graphing is induced by an isomorphism of graphings. To our knowledge, this is the first general sufficient condition in measurable combinatorics for two given graphings to be isomorphic. We then prove a full measurable Whitney theorem, namely that every weak isomorphism between graphings is implemented by a finite composition of locally finite sequences of six measurable Whitney operations. The proofs combine measurable-combinatorial techniques, applications of spanning forests from percolation theory, and graph decomposition arguments. A central ingredient is the construction and analysis of infinitely-ended leafless subforests. With these tools we also show that, between weakly $2$-connected, infinitely-ended graphings, preservation of hyperfiniteness alone forces preservation of cycles, which is of independent interest. This work further develops the limit theory of matroids recently initiated by Lovász and provides its first application in finite graph theory. We show that Whitney's rigidity theorem is robust under sublinear errors for sequences of finite graphs with uniformly bounded degrees and suitable connectivity.

math.CO↗

arXiv2025

Graphings with few circulations

In 2021, motivated by graph limit theory Lovász extended most of the theory of flows to a measure theoretic setting. Using this framework, the first author constructed $d$-regular treeings that are measurably bipartite, and have no nonzero measurable circulations, that is, flows without sources or sinks. In particular, these treeings do not admit a measurable perfect matching. In this paper, we develop tools to build $d$-regular treeings where the space of circulations is exactly $k$-dimensional for any positive integer $k$. As applications, we construct 1) a treeing with a single balanced orientation, but no Schreier decoration; 2) a treeing with a single Schreier decoration; 3) and a treeing with a proper edge $d$-coloring, but no further perfect matchings. The first answers a question raised by Lovász, as this particular balanced orientation does not decompose as a linear combination of finite cycles and infinite paths.

math.CO↗

arXiv2025

Skeletons and Spectra: Bernoulli graphings are relatively Ramanujan

The aim of this paper is to investigate the spectral theory of unimodular random graphs and graphings representing them. We prove that Bernoulli graphings are relatively Ramanujan with respect to their skeleton Markov chain. That is, the part of their spectrum that comes from the random labels falls within the appropriate Alon-Boppana bound. This result complements an example due to Frączyk of an ergodic unimodular random graph with almost sure spectral gap but non-expanding Bernoulli graphing. We also highlight connections of our work with the theory of finite random graphs. Exploiting the result of Bordenave and Collins on random lifts being relatively almost Ramanujan, we prove a strengthening of our main theorem for unimodular quasi-transitive quasi-trees.

math.PR↗

arXiv2025

Convergent sequences of combinatorial submodular setfunctions

To illustrate that the notion of convergence of submodular function sequences fits reasonably into the limit theory of graphs, we describe several classes of matroids and other submodular setfunctions for which convergence of appropriate sequences can be proved. Some of the proofs are surprisingly nontrivial.

math.CO↗

arXiv2024

The CSP Dichotomy, the Axiom of Choice, and Cyclic Polymorphisms

We study Constraint Satisfaction Problems (CSPs) in an infinite context. We show that the dichotomy between easy and hard problems -- established already in the finite case -- presents itself as the strength of the corresponding De Bruijin-Erdős-type compactness theorem over ZF. More precisely, if $\mathcal{D}$ is a structure, let $K_\mathcal{D}$ stand for the following statement: for every structure $\mathcal{X}$ if every finite substructure of $\mathcal{X}$ admits a solution to $\mathcal{D}$, then so does $\mathcal{X}$. We prove that if $\mathcal{D}$ admits no cyclic polymorphism, and thus it is NP-complete by the CSP Dichotomy Theorem, then $K_\mathcal{D}$ is equivalent to the Boolean Prime Ideal Theorem (BPI) over ZF. Conversely, we also show that if $\mathcal{D}$ admits a cyclic polymorphism, and thus it is in P, then $K_\mathcal{D}$ is strictly weaker than BPI.

math.LO↗

arXiv2024

Cycle Matroids of Graphings: From Convergence to Duality

A recent line of research has concentrated on exploring the links between analytic and combinatorial theories of submodularity, uncovering several key connections between them. In this context, Lovász initiated the study of matroids from an analytic point of view and introduced the cycle matroid of a graphing. Motivated by the limit theory of graphs, the authors introduced a form of right-convergence, called quotient-convergence, for a sequence of submodular setfunctions, leading to a notion of convergence for matroids through their rank functions. In this paper, we study the connection between local-global convergence of graphs and quotient-convergence of their cycle matroids. We characterize the exposed points of associated convex sets, forming an analytic counterpart of matroid independence- and base-polytopes. Finally, we consider dual planar graphings and show that the cycle matroid of one is the cocycle matroid of its dual if and only if the underlying graphings are hyperfinite.

math.CO↗

arXiv2024

Quotient-convergence of Submodular Setfunctions

We introduce the concept of quotient-convergence for sequences of submodular set functions, providing, among others, a new framework for the study of convergence of matroids through their rank functions. Extending the limit theory of bounded degree graphs, which analyzes graph sequences via neighborhood sampling, we address the challenge posed by the absence of a neighborhood concept in matroids. We show that any bounded set function can be approximated by a sequence of finite set functions that quotient-converges to it. In addition, we explicitly construct such sequences for increasing, submodular, and upper continuous set functions, and prove the completeness of the space under quotient-convergence.

math.CO↗

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↗

arXiv2021

Factor-of-iid Schreier decorations of lattices in Euclidean spaces

A Schreier decoration is a combinatorial coding of an action of the free group $F_d$ on the vertex set of a $2d$-regular graph. We investigate whether a Schreier decoration exists on various countably infinite transitive graphs as a factor of iid. We show that $\mathbb{Z}^d,d\geq3$, the square lattice and also the three other Archimedean lattices of even degree have finitary-factor-of-iid Schreier decorations, and exhibit examples of transitive graphs of arbitrary even degree in which obtaining such a decoration as a factor of iid is impossible. We also prove that symmetrical planar lattices with all degrees even have a factor of iid balanced orientation, meaning the indegree of every vertex is equal to its outdegree, and demonstrate that the property of having a factor-of-iid balanced orientation is not invariant under quasi-isometry.

math.PR↗

arXiv2021

Factor-of-iid balanced orientation of non-amenable graphs

We show that if a non-amenable, quasi-transitive, unimodular graph $G$ has all degrees even then it has a factor-of-iid balanced orientation, meaning each vertex has equal in- and outdegree. This result involves extending earlier spectral-theoretic results on Bernoulli shifts to the Bernoulli graphings of quasi-transitive, unimodular graphs. As a consequence, we also obtain that when $G$ is regular (of either odd or even degree) and bipartite, it has a factor-of-iid perfect matching. This generalizes a result of Lyons and Nazarov beyond transitive graphs.

math.PR↗

arXiv2021

Invariant random subgroups of groups acting on rooted trees

We investigate invariant random subgroups in groups acting on rooted trees. Let $\mathrm{Alt}_f(T)$ be the group of finitary even automorphisms of the $d$-ary rooted tree $T$. We prove that a nontrivial ergodic IRS of $\mathrm{Alt}_f(T)$ that acts without fixed points on the boundary of $T$ contains a level stabilizer, in particular it is the random conjugate of a finite index subgroup. Applying the technique to branch groups we prove that an ergodic IRS in a finitary regular branch group contains the derived subgroup of a generalized rigid level stabilizer. We also prove that every weakly branch group has continuum many distinct atomless ergodic IRS's. This extends a result of Benli, Grigorchuk and Nagnibeda who exhibit a group of intermediate growth with this property.

math.GR↗

arXiv2019

Invariant Schreier decorations of unimodular random networks

We prove that every $2d$-regular unimodular random network carries an invariant random Schreier decoration. Equivalently, it is the Schreier coset graph of an invariant random subgroup of the free group $F_d$. As a corollary we get that every $2d$-regular graphing is the local isomorphic image of a graphing coming from a p.m.p. action of $F_d$. The key ingredients of the analogous statement for finite graphs do not generalize verbatim to the measurable setting. We find a more subtle way of adapting these ingredients and prove measurable coloring theorems for graphings along the way.

math.GR↗

arXiv2017

Uniform rank gradient, cost and local-global convergence

We analyze the rank gradient of finitely generated groups with respect to sequences of subgroups of finite index that do not necessarily form a chain, by connecting it to the cost of p.m.p. actions. We generalize several results that were only known for chains before. The connection is made by the notion of local-global convergence. In particular, we show that for a finitely generated group $Γ$ with fixed price $c$, every Farber sequence has rank gradient $c-1$. By adapting Lackenby's trichotomy theorem to this setting, we also show that in a finitely presented amenable group, every sequence of subgroups with index tending to infinity has vanishing rank gradient.

math.GR↗