arXiv subjects

Matan Harel

Publications and source records attributed to Matan Harel.

16 recordsLinked to original sources

arXiv2026

Upper tails for irregular graphs beyond the mean-field regime

Let $G_{n,p}$ be the binomial random graph of density $p$ and let $X_H$ be the number of copies of a fixed graph $H$ in $G_{n,p}$. We prove asymptotically tight bounds on the logarithmic upper-tail probability of $X_H$ whenever $H$ is a connected, irregular graph with maximum degree $\Delta \ge 2$ and $p \ge n^{-1/\Delta - \varepsilon_H} (\log n)^{\omega(1)}$ for an explicit $\varepsilon_H >0$. These bounds are expressed in terms of a new variational problem that generalises the combinatorial optimisation problem arising from the na\"ive mean-field approximation. This new variational problem includes an entropy term that corresponds to the large number of embeddings of certain highly structured graphs in $K_n$. For a certain class of irregular graphs $H$ that we call stable, we show that this description of the upper-tail probability is valid in a range of densities that is optimal up to a poly($\log\log n$) factor. For a further subclass of stable graphs, which includes all irregular complete bipartite graphs, we show that this range of densities is optimal up to a multiplicative constant.

math.PR↗

arXiv2026

Planar percolation and the loop O(n) model

We show that a large class of site percolation processes on any planar graph contains either zero or infinitely many infinite connected components. The assumptions that we require are: tail triviality, positive association (FKG) and that the set of open vertices is stochastically dominated by the set of closed ones. This covers the case of Bernoulli site percolation at parameter $p\leq 1/2$ and resolves Conjecture 8 from the work of Benjamini and Schramm from 1996. Our result also implies that $p_c\geq 1/2$ for any invariantly amenable unimodular random rooted planar graph. Furthermore, we apply our statement to the loop O(n) model on the hexagonal lattice and confirm a part of the phase diagram conjectured by Nienhuis in 1982: the existence of infinitely many loops around every face whenever $n\in [1,2]$ and $x\in [1/\sqrt{2},1]$. The point $n=2,x=1/\sqrt{2}$ is conjectured to be critical. This is the first instance that this behavior has been proven in such a large region of parameters. In a big portion of this region, the loop O(n) model has no known FKG representation. We apply our percolation result to quenched distributions that can be described as divide and color models.

math.PR↗

arXiv2024

Upper tails for arithmetic progressions revisited

Let $X$ be the number of $k$-term arithmetic progressions contained in the $p$-biased random subset of the first $N$ positive integers. We give asymptotically sharp estimates on the logarithmic upper-tail probability $\log \Pr(X \ge E[X] + t)$ for all $Ω(N^{-2/k}) \le p \ll 1$ and all $t \gg \sqrt{Var(X)}$, excluding only a few boundary cases. In particular, we show that the space of parameters $(p,t)$ is partitioned into three phenomenologically distinct regions, where the upper-tail probabilities either resemble those of Gaussian or Poisson random variables, or are naturally described by the probability of appearance of a small set that contains nearly all of the excess $t$ progressions. We employ a variety of tools from probability theory, including classical tilting arguments and martingale concentration inequalities. However, the main technical innovation is a combinatorial result that establishes a stronger version of `entropic stability' for sets with rich arithmetic structure.

math.PR↗

arXiv2024

Quantitative disorder effects in low-dimensional spin systems

The Imry-Ma phenomenon, predicted in 1975 by Imry and Ma and rigorously established in 1989 by Aizenman and Wehr, states that first-order phase transitions of low-dimensional spin systems are `rounded' by the addition of a quenched random field to the quantity undergoing the transition. The phenomenon applies to a wide class of spin systems in dimensions $d\le 2$ and to spin systems possessing a continuous symmetry in dimensions $d\le 4$. This work provides quantitative estimates for the Imry--Ma phenomenon: In a cubic domain of side length $L$, we study the effect of the boundary conditions on the spatial and thermal average of the quantity coupled to the random field. We show that the boundary effect diminishes at least as fast as an inverse power of $\log\log L$ for general two-dimensional spin systems and for four-dimensional spin systems with continuous symmetry, and at least as fast as an inverse power of $L$ for two- and three-dimensional spin systems with continuous symmetry. Specific models of interest for the obtained results include the two-dimensional random-field $q$-state Potts and Edwards-Anderson spin glass models, and the $d$-dimensional random-field spin $O(n)$ models ($n\ge 2$) in dimensions $d\le 4$.

math-ph↗

arXiv2024

Macroscopic loops in the loop O(n) model via the XOR trick

The loop $O(n)$ model is a family of probability measures on collections of non-intersecting loops on the hexagonal lattice, parameterized by a loop-weight $n$ and an edge-weight $x$. Nienhuis predicts that, for $0 \leq n \leq 2$, the model exhibits two regimes separated by $x_c(n) = 1/\sqrt{2 + \sqrt{2-n}}$: when $x < x_c(n)$, the loop lengths have exponential tails, while, when $x \geq x_c(n)$, the loops are macroscopic. In this paper, we prove three results regarding the existence of long loops in the loop $O(n)$ model: - In the regime $(n,x) \in [1,1+δ) \times (1- δ, 1]$ with $δ>0$ small, a configuration sampled from a translation-invariant Gibbs measure will either contain an infinite path or have infinitely many loops surrounding every face. In the subregime $n \in [1,1+δ)$ and $x \in (1-δ,1/\sqrt{n}]$ our results further imply Russo--Seymour--Welsh theory. This is the first proof of the existence of macroscopic loops in a positive area subset of the phase diagram. - Existence of loops whose diameter is comparable to that of a finite domain whenever $n=1, x \in (1,\sqrt{3}]$; this regime is equivalent to part of the antiferromagnetic regime of the Ising model on the triangular lattice. - Existence of non-contractible loops on a torus when $n \in [1,2], x=1$. The main ingredients of the proof are: (i) the `XOR trick': if $ω$ is a collection of short loops and $Γ$ is a long loop, then the symmetric difference of $ω$ and $Γ$ necessarily includes a long loop as well; (ii) a reduction of the problem of finding long loops to proving that a percolation process on an auxiliary planar graph, built using the Chayes--Machta and Edwards--Sokal geometric expansions, has no infinite connected components; and (iii) a recent result on the percolation threshold of Benjamini--Schramm limits of planar graphs.

math.PR↗

arXiv2023

Random-field random surfaces

We study how the typical gradient and typical height of a random surface are modified by the addition of quenched disorder in the form of a random independent external field. The results provide quantitative estimates, sharp up to multiplicative constants, in the following cases. It is shown that for real-valued disordered random surfaces of the $\nabla ϕ$ type with a uniformly convex interaction potential: (i) The gradient of the surface delocalizes in dimensions $1\le d\le 2$ and localizes in dimensions $d\ge3$. (ii) The surface delocalizes in dimensions $1\le d\le 4$ and localizes in dimensions $d\ge 5$. It is further shown that for the integer-valued disordered Gaussian free field: (i) The gradient of the surface delocalizes in dimensions $d=1,2$ and localizes in dimensions $d\ge3$. (ii) The surface delocalizes in dimensions $d=1,2$. (iii) The surface localizes in dimensions $d\ge 3$ at weak disorder strength. The behavior in dimensions $d\ge 3$ at strong disorder is left open. The proofs rely on several tools: explicit identities satisfied by the expectation of the random surface, the Efron--Stein concentration inequality, a coupling argument for Langevin dynamics (originally due to Funaki and Spohn) and the Nash--Aronson estimate.

math-ph↗

arXiv2022

Depinning in integer-restricted Gaussian Fields and BKT phases of two-component spin models

For a family of integer-valued height functions defined over the faces of planar graphs, we establish a relation between the probability of connection by level sets and the spin-spin correlations of the dual $O(2)$ symmetric spin models formulated over the graphs' vertices. The relation is used to show that in two dimensions the Villain spin model exhibits non-summable decay of correlations at any temperature at which the dual integer-restricted Gaussian field exhibits depinning. For the latter, we devise a new monotonicity argument through which the recent alternative proof by Lammers of the existence of a depinning transition in two-dimensional graphs of degree three, is extended to all doubly-periodic graphs, in particular to $\mathbb{Z}^2$. Essential use is made of the inequality of Regev and Stephens-Davidowitz, which allows also an alternative (to absolute-value FKG) proof of convergence of the height-function's distribution in the infinite-volume limit. Similar results are established for the $XY$ spin model and its dual Bessel random height function. Taken together these statements yield a new perspective on the Berezinskii-Kosterlitz-Thouless phase transition in $O(2)$ spin models, and complete a new proof of depinning in two-dimensional integer-valued height functions.

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

Logarithmic variance for the height function of square-ice

In this article, we prove that the height function associated with the square-ice model (i.e.~the six-vertex model with $a=b=c=1$ on the square lattice), or, equivalently, of the uniform random homomorphisms from $\mathbb Z^2$ to $\mathbb Z$, has logarithmic variance. This establishes a strong form of roughness of this height function.

math.PR↗

arXiv2021

Upper tails via high moments and entropic stability

Suppose that $X$ is a bounded-degree polynomial with nonnegative coefficients on the $p$-biased discrete hypercube. Our main result gives sharp estimates on the logarithmic upper tail probability of $X$ whenever an associated extremal problem satisfies a certain entropic stability property. We apply this result to solve two long-standing open problems in probabilistic combinatorics: the upper tail problem for the number of arithmetic progressions of a fixed length in the $p$-random subset of the integers and the upper tail problem for the number of cliques of a fixed size in the random graph $G_{n,p}$. We also make significant progress on the upper tail problem for the number of copies of a fixed regular graph $H$ in $G_{n,p}$. To accommodate readers who are interested in learning the basic method, we include a short, self-contained solution to the upper tail problem for the number of triangles in $G_{n,p}$ for all $p=p(n)$ satisfying $n^{-1}\log n\ll p \ll 1$.

math.PR↗

arXiv2020

Rational Groupthink

We study how long-lived rational agents learn from repeatedly observing a private signal and each others' actions. With normal signals, a group of any size learns more slowly than just four agents who directly observe each others' private signals in each period. Similar results apply to general signal structures. We identify rational groupthink---in which agents ignore their private signals and choose the same action for long periods of time---as the cause of this failure of information aggregation.

cs.GT↗

arXiv2019

Exponential decay of correlations in the 2D random field Ising model

An extension of the Ising spin configurations to continuous functions is used for an exact representation of the Random Field Ising Model's order parameter in terms of disagreement percolation. This facilitates an extension of the recent analyses of the decay of correlations to positive temperatures, at homogeneous but arbitrarily weak disorder.

math-ph↗

arXiv2019

Localization in random geometric graphs with too many edges

We consider a random geometric graph $G(χ_n, r_n)$, given by connecting two vertices of a Poisson point process $χ_n$ of intensity $n$ on the unit torus whenever their distance is smaller than the parameter $r_n$. The model is conditioned on the rare event that the number of edges observed, $|E|$, is greater than $(1 + δ)\mathbb{E}(|E|)$, for some fixed $δ>0$. This article proves that upon conditioning, with high probability there exists a ball of diameter $r_n$ which contains a clique of at least $\sqrt{2 δ\mathbb{E}(|E|)}(1 - \varepsilon)$ vertices, for any given $\varepsilon >0$. Intuitively, this region contains all the "excess" edges the graph is forced to contain by the conditioning event, up to lower order corrections. As a consequence of this result, we prove a large deviations principle for the upper tail of the edge count of the random geometric graph. The rate function of this large deviation principle turns out to be non-convex.

math.PR↗

arXiv2017

Discontinuity of the phase transition for the planar random-cluster and Potts models with $q>4$

We prove that the $q$-state Potts model and the random-cluster model with cluster weight $q>4$ undergo a discontinuous phase transition on the square lattice. More precisely, we show - Existence of multiple infinite-volume measures for the critical Potts and random-cluster models, - Ordering for the measures with monochromatic (resp. wired) boundary conditions for the critical Potts model (resp. random-cluster model), and - Exponential decay of correlations for the measure with free boundary conditions for both the critical Potts and random-cluster models. The proof is based on a rigorous computation of the Perron-Frobenius eigenvalues of the diagonal blocks of the transfer matrix of the six-vertex model, whose ratios are then related to the correlation length of the random-cluster model. As a byproduct, we rigorously compute the correlation lengths of the critical random-cluster and Potts models, and show that they behave as $\exp(π^2/\sqrt{q-4})$ as $q$ tends to 4.

math.PR↗

arXiv2016

The Bethe ansatz for the six-vertex and XXZ models: an exposition

In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector $ψ$ and energy $Λ$, which satisfy $V ψ= Λψ$, where $V$ is the the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights $a= b=1$ and $c > 0$. We also show that the same vector $ψ$ satisfies $H ψ= E ψ$, where $H$ is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value $E$ computed explicitly. Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors that amounts to proving that the random cluster model on $\mathbb{Z}^2$ with cluster weight $q >4$ exhibits a first-order phase transition.

math.PR↗

arXiv2011

Affine and Projective Tree Metric Theorems

The tree metric theorem provides a combinatorial four point condition that characterizes dissimilarity maps derived from pairwise compatible split systems. A similar (but weaker) four point condition characterizes dissimilarity maps derived from circular split systems (Kalmanson metrics). The tree metric theorem was first discovered in the context of phylogenetics and forms the basis of many tree reconstruction algorithms, whereas Kalmanson metrics were first considered by computer scientists, and are notable in that they are a non-trivial class of metrics for which the traveling salesman problem is tractable. We present a unifying framework for these theorems based on combinatorial structures that are used for graph planarity testing. These are (projective) PC-trees, and their affine analogs, PQ-trees. In the projective case, we generalize a number of concepts from clustering theory, including hierarchies, pyramids, ultrametrics and Robinsonian matrices, and the theorems that relate them. As with tree metrics and ultrametrics, the link between PC-trees and PQ-trees is established via the Gromov product.

math.CO↗