arXiv subjects

Yinon Spinka

Publications and source records attributed to Yinon Spinka.

At least 19 recordsLinked to original sources

arXiv2026

Random homomorphisms and Lipschitz functions on trees

A graph homomorphism is an integer-valued function on the vertex set of a graph that assigns values differing by exactly one to adjacent vertices. We consider uniformly random homomorphisms on general finite trees, conditioned to take the value zero at all leaves, and study the distribution of the value at the root. Our main result is a stochastic comparison, both from above and below, between the absolute values of the homomorphism value at the root and certain discrete Gaussian-like random variables. In particular, we obtain a subgaussian tail bound valid for all deviations, a matching lower bound that holds up to a certain threshold, and upper and lower variance bounds that differ by a constant factor. These bounds depend solely on the effective resistance between the root and the leaves in the associated electrical network. As a consequence, in the setting of infinite locally finite trees, we obtain that the homomorphism model is localized on transient trees and delocalized on recurrent trees. Analogous results are obtained for random integer-valued Lipschitz functions. Our results extend previous results of Benjamini--H\"aggstr\"om--Mossel on homomorphisms on regular trees, of Peled--Samotij--Yehudayoff on Lipschitz functions on regular trees, and of Lammers--Toninelli on homomorphisms on trees of minimum degree at least three.

math.PR↗

arXiv2026

A noisy min-max game on trees

We study a noisy version of a min-max type zero-sum game on the $d$-ary tree. Each edge of the tree is assigned an i.i.d.\ cookie, distributed uniformly on $\{+1,-1\}$. The game is played as follows: starting at the root, two players alternate turns in choosing a child to move to, with the game ending after each player took $n$ turns. Both players have full knowledge of the cookies on the whole tree. The cookies along the traversed edges are picked up and placed in a shared cookie jar. The first player's payoff is the sum of the cookies in the cookie jar, while the second player pays that sum. The value $V_n$ of the $n$-round game is the largest signed sum which can be guaranteed by the first player. We analyze the value $V_n$ and show that as $n \to \infty$, the value is tight for $d=2$, converges in distribution for $d \ge 3$, and converges almost surely for $d \ge 15$. Along the way, we prove various tightness and double exponential tail decay results. The analysis is a mix of percolation-type arguments for large $d$, and iterations on distributions combined with interval arithmetic for small $d$. For $d=2$ we prove the existence of a continuum of fixed points for this iteration, highlighting surprising qualitative differences with the case $d \ge 3$. The question of convergence for $d=2$ remains open.

math.PR↗

arXiv2026

Optimal factor matchings for point processes on non-amenable unimodular graphs

Consider a unit-intensity point process $\Pi$ on the vertex set $V$ of a transitive non-amenable unimodular graph. We study invariant matchings between $\Pi$ and $V$ having small typical matching distances. When $\Pi$ is either a Poisson process or i.i.d. perturbations of the vertex set, we determine the optimal matching distance and show that it can be attained by a factor matching scheme (that is, a deterministic and equivariant function of $\Pi$).

math.PR↗

arXiv2025

Optimal matchings of randomly perturbed lattices

Consider a point process in Euclidean space obtained by perturbing the integer lattice with independent and identically distributed random vectors. Under mild assumptions on the law of the perturbations, we construct a translation-invariant perfect matching between this point process and the lattice, such that the matching distance has the same tail behavior as the hole probability of the point process, which is a natural lower bound.

math.PR↗

arXiv2025

Finitary codings and stochastic domination for Poisson representable processes

Construct a random set by independently selecting each finite subset of the integers with some probability depending on the set up to translations and taking the union of the selected sets. We show that when the only sets selected with positive probability are pairs, such a random set is a finitary factor of an IID process, answering a question of Forsstr\"om, Gantert and Steif. More generally, we show that this is the case whenever the distribution induced by the size of the selected sets has sufficient exponential moments, and that the existence of some exponential moment is necessary. We further show that such a random set is stochastically dominated by a non-trivial Bernoulli percolation if and only if there is a finite exponential moment, thereby partially answering another question of Forsstr\"om et al. We also give a partial answer to a third question regarding a form of phase transition. These results also hold on $\mathbb{Z}^d$ with $d \ge 2$. In the one-dimensional case, under the condition that the distribution induced by the diameter of the selected sets has an exponential moment, we further show that such a random set is finitarily isomorphic to an IID process.

math.PR↗

arXiv2025

A new proof of finitary isomorphism for Markov chains

We give a new proof of a result of Rudolph stating that a countable-state mixing Markov chain with exponential return times is finitarily isomorphic to an IID process. Besides being short and direct, our proof has the added benefit of working for processes of finite or infinite entropy.

math.PR↗

arXiv2024

On the local convergence of integer-valued Lipschitz functions on regular trees

We study random integer-valued Lipschitz functions on regular trees. It was shown by Peled, Samotij and Yehudayoff that such functions are localized, however, finer questions about the structure of Gibbs measures remain unanswered. Our main result is that the weak limit of a uniformly chosen 1-Lipschitz function with 0 boundary condition on a $d$-ary tree of height $n$ exists as $n \to \infty$ if $2 \le d \le 7$, but not if $d \ge 8$, thereby partially answering a question posed by Peled, Samotij and Yehudayoff. For large $d$, the value at the root alternates between being almost entirely concentrated on 0 for even $n$ and being roughly uniform on $\{-1,0,1\}$ for odd $n$, leading to different limits as $n$ approaches infinity along evens or odds. For $d \ge 8$, the essence of this phenomenon is preserved, which obstructs the convergence. For $d \le 7$, this phenomenon ceases to exist, and the law of the value at the root loses its connection with the parity of $n$. Along the way, we also obtain an alternative proof of localization. The key idea is a fixed point convergence result for a related operator on $\ell^\infty$, and a procedure to show that the iterations get into a `basin of attraction' of the fixed point. We also prove some accompanying analogous `even-odd phenomenon' type results about $M$-lipschitz functions on general non-amenable graphs with high enough expansion (this includes for example the large $d$ case for regular trees). We also prove a convergence result for 1-Lipschitz functions with $\{0,1\}$ boundary condition. This last result relies on an absolute value FKG for uniform 1-Lipschitz functions when shifted by $1/2$.

math.PR↗

arXiv2023

Characterizations of amenability through stochastic domination and finitary codings

We establish new characterizations of amenability of graphs through two probabilistic notions: stochastic domination and finitary codings (also called finitary factors). On the stochastic domination side, we show that the plus state of the Ising model at very low temperature stochastically dominates a high density Bernoulli percolation if and only if the underlying graph is nonamenable. This answers a question of Liggett and Steif. We prove a similar result for the ``infinite cluster process'' of Bernoulli percolation, which is of particular interest as this process is not monotone and does not possess any nice form of the domain Markov property. We also prove that the plus states of the Ising model at very low temperatures are stochastically ordered if and only if the graph is nonamenable. This answers a second question of Liggett and Steif. We further show that these stochastic domination results can be witnessed by invariant monotone couplings. On the finitary coding side, we show that the plus state of the Ising model at very low temperature is a finitary factor of an i.i.d. process if and only if the underlying graph is nonamenable (assuming it supports a phase transition). We show a similar result for the infinite cluster process of a high-density Bernoulli percolation. A main technique is to consider diluted processes, for which Holley's condition can be established. In order to compare two Ising models, we apply a dilution mechanism using lattice gas theory. Along the way we develop general tools to establish invariant domination when Holley's condition is satisfied. The finitary factor results are based on the stochastic domination results and a dynamical construction involving bounding chains, along with a new technique to analyze coupling from the past in infinite-range processes via a novel disease spreading model, which may be of independent interest.

math.PR↗

arXiv2023

Random walks on regular trees can not be slowed down

A random walk on a regular tree (or any non-amenable graph) has positive speed. We ask whether such a walk can be slowed down by applying carefully chosen time-dependent permutations of the vertices. We prove that on trees the random walk can not be slowed down.

math.PR↗

arXiv2022

Three lectures on random proper colorings of $\mathbb{Z}^d$

A proper $q$-coloring of a graph is an assignment of one of $q$ colors to each vertex of the graph so that adjacent vertices are colored differently. Sample uniformly among all proper $q$-colorings of a large discrete cube in the integer lattice $\mathbb{Z}^d$. Does the random coloring obtained exhibit any large-scale structure? Does it have fast decay of correlations? We discuss these questions and the way their answers depend on the dimension $d$ and the number of colors $q$. The questions are motivated by statistical physics (anti-ferromagnetic materials, square ice), combinatorics (proper colorings, independent sets) and the study of random Lipschitz functions on a lattice. The discussion introduces a diverse set of tools, useful for this purpose and for other problems, including spatial mixing, entropy and coupling methods, Gibbs measures and their classification and refined contour analysis.

math.PR↗

arXiv2022

Finitary codings for the random-cluster model and other infinite-range monotone models

A random field $X = (X_v)_{v \in G}$ on a quasi-transitive graph $G$ is a factor of i.i.d. if it can be written as $X=φ(Y)$ for some i.i.d. process $Y= (Y_v)_{v \in G}$ and equivariant map $φ$. Such a map, also called a coding, is finitary if, for every vertex $v \in G$, there exists a finite (but random) set $U \subset G$ such that $X_v$ is determined by $\{Y_u\}_{u \in U}$. We construct a coding for the random-cluster model on $G$, and show that the coding is finitary whenever the free and wired measures coincide. This strengthens a result of Häggström--Jonasson--Lyons. We also prove that the coding radius has exponential tails in the subcritical regime. As a corollary, we obtain a similar coding for the subcritical Potts model. Our methods are probabilistic in nature, and at their heart lies the use of coupling-from-the-past for the Glauber dynamics. These methods apply to any monotone model satisfying mild technical (but natural) requirements. Beyond the random-cluster and Potts models, we describe two further applications -- the loop $O(n)$ model and long-range Ising models. In the case of $G = \mathbb{Z}^d$, we also construct finitary, translation-equivariant codings using a finite-valued i.i.d. process $Y$. To do this, we extend a mixing-time result of Martinelli--Olivieri to infinite-range monotone models on quasi-transitive graphs of sub-exponential growth.

math.PR↗

arXiv2022

Finitary codings for gradient models and a new graphical representation for the six-vertex model

It is known that the Ising model on $\mathbb {Z}^d$ at a given temperature is a finitary factor of an i.i.d. process if and only if the temperature is at least the critical temperature. Below the critical temperature, the plus and minus states of the Ising model are distinct and differ from one another by a global flip of the spins. We show that it is only this global information which poses an obstruction for being finitary by showing that the gradient of the Ising model is a finitary factor of i.i.d. at all temperatures. As a consequence, we deduce a volume-order large deviation estimate for the energy. A similar result is shown for the Potts model. A result in the same spirit is also shown for the six-vertex model, which is itself the gradient of a height function, with parameter $c \gtrapprox 6.4$. We show that the gradient of the height function is not a finitary factor of an i.i.d. process, but that its "Laplacian" is. For this, we introduce a coupling between the six-vertex model with $c\ge 2$ and a new graphical representation of it, reminiscent of the Edwards--Sokal coupling between the Potts and random-cluster models. We believe that this graphical representation may be of independent interest and could serve as a tool in further understanding of the six-vertex model. To provide further support for the ubiquity of this type of phenomenon, we also prove an analogous result for the so-called beach model. The tools and techniques used in this paper are probabilistic in nature. The heart of the argument is to devise a suitable tree structure on the clusters of the underlying percolation process (associated to the graphical representation of the given model), which can be revealed piece-by-piece via exploration.

math.PR↗

arXiv2022

Entropy-efficient finitary codings

We show that any finite-entropy, countable-valued finitary factor of an i.i.d process can also be expressed as a finitary factor of a finite-valued i.i.d process whose entropy is arbitrarily close to the target process. As an application, we give an affirmative answer to a question of van den Berg and Steif about the critical Ising model on $\mathbb{Z}^d$. En route, we prove several results about finitary isomorphisms and finitary factors. Our results are developed in a new framework for processes invariant to a permutation group of a countable set satisfying specific properties. This new framework includes all ``classical'' processes over countable amenable groups and all invariant processes on transitive amenable graphs with ``uniquely centered balls''. Some of our results are new already for $\mathbb{Z}$-processes. We prove a relative version of Smorodinsky's isomorphism theorem for finitely dependent $\mathbb{Z}$-processes. We also extend the Keane--Smorodinsky finitary isomorphism theorem to countable-valued i.i.d processes and to i.i.d processes taking values in a Polish space.

math.PR↗

arXiv2022

Independent sets in random subgraphs of the hypercube

Let $Q_{d,p}$ be the random subgraph of the $d$-dimensional hypercube $\{0,1\}^d$, where each edge is retained independently with probability $p$. We study the asymptotic number of independent sets in $Q_{d,p}$ as $d \to \infty$ for a wide range of parameters $p$, including values of $p$ tending to zero as fast as $\frac{C\log d}{d^{1/3}}$, constant values of $p$, and values of $p$ tending to one. The results extend to the hardcore model on $Q_{d,p}$, and are obtained by studying the closely related antiferromagnetic Ising model on the hypercube, which can be viewed as a positive-temperature hardcore model on the hypercube. These results generalize previous results by Galvin, Jenssen and Perkins on the hard-core model on the hypercube, corresponding to the case $p=1$, which extended Korshunov and Sapozhenko's classical result on the asymptotic number of independent sets in the hypercube.

math.CO↗

arXiv2021

Uniform even subgraphs and graphical representations of Ising as factors of i.i.d

We prove that the Loop O(1) model, a well-known graphical expansion of the Ising model, is a factor of i.i.d. on unimodular random rooted graphs under various conditions, including in the presence of a non-negative external field. As an application we show that the gradient of the free Ising model is a factor of i.i.d. on unimodular planar maps having a locally finite dual. The key idea is to develop an appropriate theory of local limits of uniform even subgraphs with various boundary conditions and prove that they can be sampled as a factor of i.i.d. Another key tool we prove and exploit is that the wired uniform spanning tree on a unimodular transient graph is a factor of i.i.d. This partially answers some questions posed by Hutchcroft.

math.PR↗

arXiv2021

Geometric random graphs on circles

Given a dense countable set in a metric space, the infinite random geometric graph is the random graph with the given vertex set and where any two points at distance less than 1 are connected, independently, with some fixed probability. It has been observed by Bonato and Janssen that in some, but not all, such settings, the resulting graph does not depend on the random choices, in the sense that it is almost surely isomorphic to a fixed graph. While this notion makes sense in the general context of metric spaces, previous work has been restricted to sets in Banach spaces. We study the case when the underlying metric space is a circle of circumference $L$, and find a surprising dependency of behavior on the rationality of $L$.

math.CO↗

arXiv2021

Pairwise optimal coupling of multiple random variables

We generalize the optimal coupling theorem to multiple random variables: Given a collection of random variables, it is possible to couple all of them so that any two differ with probability comparable to the total-variation distance between them. In a number of cases we show that the disagreement probability we achieve is the best possible. The proofs of sharpness rely on new results in extremal combinatorics, which may be of independent interest.

math.PR↗

arXiv2021

A tale of two balloons

From each point of a Poisson point process start growing a balloon at rate 1. When two balloons touch, they pop and disappear. Is every point contained in balloons infinitely often or not? We answer this for the Euclidean space, the hyperbolic plane and regular trees. The result for the Euclidean space relies on a novel 0-1 law for stationary processes. Towards establishing the results for the hyperbolic plane and regular trees, we prove an upper bound on the density of any well-separated set in a regular tree which is a factor of an i.i.d. process.

math.PR↗