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Ron Peled

Publications and source records attributed to Ron Peled.

At least 19 recordsLinked to original sources
arXiv2026

Entropic repulsion to the middle layer

We consider $\nabla\varphi$ height functions with even and convex interaction energy $W$ on the lattice $\mathbb{Z}^d$, which are restricted to take values in the set $\{-S, \ldots, S\}$ for some integer $S \ge 1$. We study the effect of entropic repulsion, which tends to push the spin values to the middle layer. We prove that the model has a unique Gibbs measure, with exponential decay of correlations, in the cases: (a) Dimension $d=2$ at all temperatures. (b) Dimensions $d\ge3$ at all temperatures, for a wide class of $W$ with non-increasing second derivative, including the family $W(x)=|x|^p$ for $p\in[1,2]$. (c) Dimensions $d\ge 3$ at both low and high temperatures, $T\in (0,\frac{W(1)d}{4(\ln d+\ln 8)})\cup(d W(2S),\infty]$, with the normalization $W(0)=0$. At low temperatures, our proof provides an alternative to Pirogov--Sinai methods. Conversely, we exhibit a class of even and convex interaction energies $W$ which, in high dimensions and suitable temperature regimes, have multiple Gibbs measures. Though uniqueness may fail, we show that the magnetization of every Gibbs measure lies in $(-\frac{1}{2},\frac{1}{2})$. This implies the delocalization of the model restricted to take values in $\{0,1,\ldots\}$ (i.e., conditioned to lie above a floor) for all dimensions, any even and convex $W$, and all temperatures. Our methods extend to additional setups: We prove that height functions taking values in the real interval $[-1,1]$ always have a unique Gibbs measure, a result previously proved only for the quadratic interaction. For height functions taking values in $\{-S+\frac{1}{2}, \ldots, S-\frac{1}{2}\}$, $S \ge1 $ integer, we prove that the magnetization of every Gibbs measure lies in $(-1,1)$. The special case $W(x)=x^2$ of our results addresses questions left open in the work of Bricmont--El Mellouki--Fr\"ohlich (1986).

math-ph↗
arXiv2026

On the rigidity of sloped height functions in $d\ge 3$ and non-crossing surfaces

We consider integer-valued $\nabla\phi$ height functions, with general convex interactions, placed on a slope. Sheffield (2003) conjectured that for all slopes, such height functions are localized in dimensions $d\ge3$, in the sense of having tight fluctuations in finite volume, and admitting infinite-volume limits. We establish this conjecture when there are two coordinates on which the slope vector is zero and on which the interactions are even and low temperature. In this setting, we also prove the uniqueness, in the appropriate sense, of the infinite-volume limit and classify its extremal components. We further study the structural properties of the infinite-volume limit. We establish the entropic repulsion between macroscopic domain walls (boundaries of level sets), showing that they are maximally separated in a precise sense described by a rotation-of-the-circle dynamical system. Lastly, we show exponential decay of correlations in the extremal components. Our setup includes, as a special case, infinitely many zero-slope integer-valued surfaces (of dimension two or higher) conditioned not to cross. We deduce the existence and structural properties of the bulk Gibbs measure over such non-crossing surfaces with any given average spacing.

math.PR↗
arXiv2026

On the critical fugacity of the hard-core model on regular bipartite graphs

We establish long-range order for the hard-core model on a finite, regular bipartite graph above a threshold fugacity given in terms of expansion parameters of the graph. The result applies to the $d$-dimensional hypercube graph and, more generally, to $d$-dimensional discrete tori of fixed side length, proving long-range order at fugacities $\lambda\ge\Omega(\frac{\log d}{d})$. Furthermore, we use reflection positivity to transfer the result to the lattice $\mathbb{Z}^{d}$, verifying the long-standing belief that its critical fugacity is of the form $d^{-1+o(1)}$ as $d\to\infty$.

math.PR↗
arXiv2025

Liquid-vapor transition in a model of a continuum particle system with finite-range modified Kac pair potential

We prove the existence of a phase transition in dimension $d>1$ in a continuum particle system interacting with a pair potential containing a modified attractive Kac potential of range $\gamma^{-1}$, with $\gamma>0$. This transition is "close", for small positive $\gamma$, to the one proved previously by Lebowitz and Penrose in the van der Waals limit $\gamma\downarrow0$. It is of the type of the liquid-vapor transition observed when a fluid, like water, heated at constant pressure, boils at a given temperature. Previous results on phase transitions in continuum systems with stable potentials required the use of unphysical four-body interactions or special symmetries between the liquid and vapor. The pair interaction we consider is obtained by partitioning space into cubes of volume $\gamma^{-d}$, and letting the Kac part of the pair potential be uniform in each cube and act only between adjacent cubes. The "short-range" part of the pair potential is quite general (in particular, it may or may not include a hard core), but restricted to act only between particles in the same cube. Our setup, the "boxed particle model", is a special case of a general "spin" system, for which we establish a first-order phase transition using reflection positivity and the Dobrushin--Shlosman criterion.

math-ph↗
arXiv2025

Minimal surfaces in strongly correlated random environments

A minimal surface in a random environment (MSRE) is a $d$-dimensional surface in $(d+n)$-dimensional space which minimizes the sum of its elastic energy and its environment potential energy, subject to prescribed boundary values. Apart from their intrinsic interest, such surfaces are further motivated by connections with disordered spin systems and first-passage percolation models. In this work, we consider the case of strongly correlated environments, realized by the model of harmonic MSRE in a fractional Brownian environment of Hurst parameter $H\in(0,1)$. This includes the case of Brownian environment ($H=1/2$ and $n=1$), which is commonly used to approximate the domain walls of the $(d+1)$-dimensional random-field Ising model. We prove that surfaces of dimension $d\in\{1,2,3\}$ delocalize with power-law fluctuations, and determine their precise transversal and minimal energy fluctuation exponents, as well as the stretched exponential exponents governing the tail decay of their distributions. These exponents are found to be the same in all codimensions $n$, depending only on $d$ and $H$. The transversal and minimal energy fluctuation exponents are specified by two scaling relations. We further show that surfaces of dimension $d=4$ delocalize with sub-power-law fluctuations, with their height and minimal energy fluctuations tied by a scaling relation. Lastly, we prove that surfaces of dimensions $d\ge 5$ localize. These results put several predictions from the physics literature on mathematically rigorous ground.

math.PR↗
arXiv2024

Minimal surfaces in random environment

A minimal surface in a random environment (MSRE) is a surface which minimizes the sum of its elastic energy and its environment potential energy, subject to prescribed boundary conditions. Apart from their intrinsic interest, such surfaces are further motivated by connections with disordered spin systems and first-passage percolation models. We wish to study the geometry of $d$-dimensional minimal surfaces in a $(d+n)$-dimensional random environment. Specializing to a model that we term harmonic MSRE, in an ``independent'' random environment, we rigorously establish bounds on the geometric and energetic fluctuations of the minimal surface, as well as versions of the scaling relation $\chi=2\xi+d-2$ that ties together these two types of fluctuations. In particular, we prove, for all values of $n$, that the surfaces are delocalized in dimensions $d\le 4$ and localized in dimensions $d\ge 5$. Moreover, the surface delocalizes with power-law fluctuations when $d\le 3$ and sub-power-law fluctuations when $d=4$. Our localization results apply also to harmonic minimal surfaces in a periodic random environment.

math-ph↗
arXiv2023

Non-constant ground configurations in the disordered ferromagnet

The disordered ferromagnet is a disordered version of the ferromagnetic Ising model in which the coupling constants are non-negative quenched random. A ground configuration is an infinite-volume configuration whose energy cannot be reduced by finite modifications. It is a long-standing challenge to ascertain whether the disordered ferromagnet on the $\mathbb{Z}^D$ lattice admits non-constant ground configurations. We answer this affirmatively in dimensions $D\ge 4$, when the coupling constants are sampled independently from a sufficiently concentrated distribution. The obtained ground configurations are further shown to be translation-covariant with respect to $\mathbb{Z}^{D-1}$ translations of the disorder. Our result is proved by showing that the finite-volume interface formed by Dobrushin boundary conditions is localized, and converges to an infinite-volume interface. This may be expressed in purely combinatorial terms, as a result on the fluctuations of certain minimal cutsets in the lattice $\mathbb{Z}^D$ endowed with independent edge capacities.

math-ph↗
arXiv2023

On the influence of edges in first-passage percolation on $\mathbb{Z}^d$

We study first-passage percolation on $\mathbb Z^d$, $d\ge 2$, with independent weights whose common distribution is compactly supported in $(0,\infty)$ with a uniformly-positive density. Given $\epsilon>0$ and $v\in\mathbb Z^d$, which edges have probability at least $\epsilon$ to lie on the geodesic between the origin and $v$? It is expected that all such edges lie at distance at most some $r(\epsilon)$ from either the origin or $v$, but this remains open in dimensions $d\ge 3$. We establish the closely-related fact that the number of such edges is at most some $C(\epsilon)$, uniformly in $v$. In addition, we prove a quantitative bound, allowing $\epsilon$ to tend to zero as $\|v\|$ tends to infinity, showing that there are at most $O\big(\epsilon^{-\frac{2d}{d-1}}(\log \|v\|)^C\big)$ such edges, uniformly in $\epsilon$ and $v$. The latter result addresses a problem raised by Benjamin-Kalai-Schramm (2003). Our technique further yields a strengthened version of a lower bound on transversal fluctuations due to Licea-Newman-Piza (1996).

math.PR↗
arXiv2022

Columnar order in random packings of $2\times2$ squares on the square lattice

We study random packings of $2\times2$ squares with centers on the square lattice $\mathbb{Z}^{2}$, in which the probability of a packing is proportional to $\lambda$ to the number of squares. We prove that for large $\lambda$, typical packings exhibit columnar order, in which either the centers of most tiles agree on the parity of their $x$-coordinate or the centers of most tiles agree on the parity of their $y$-coordinate. This manifests in the existence of four extremal and periodic Gibbs measures in which the rotational symmetry of the lattice is broken while the translational symmetry is only broken along a single axis. We further quantify the decay of correlations in these measures, obtaining a slow rate of exponential decay in the direction of preserved translational symmetry and a fast rate in the direction of broken translational symmetry. Lastly, we prove that every periodic Gibbs measure is a mixture of these four measures. Additionally, our proof introduces an apparently novel extension of the chessboard estimate, from finite-volume torus measures to all infinite-volume periodic Gibbs measures.

math-ph↗
arXiv2022

Coalescence of geodesics and the BKS midpoint problem in planar first-passage percolation

We consider first-passage percolation on $\mathbb Z^2$ with independent and identically distributed weights whose common distribution is absolutely continuous with a finite exponential moment. Under the assumption that the limit shape has more than 32 extreme points, we prove that geodesics with nearby starting and ending points have significant overlap, coalescing on all but small portions near their endpoints. The statement is quantified, with power-law dependence of the involved quantities on the length of the geodesics. The result leads to a quantitative resolution of the Benjamini--Kalai--Schramm midpoint problem. It is shown that the probability that the geodesic between two given points passes through a given edge is smaller than a power of the distance between the points and the edge. We further prove that the limit shape assumption is satisfied for a specific family of distributions. Lastly, related to the 1965 Hammersley--Welsh highways and byways problem, we prove that the expected fraction of the square $\{-n,\dots ,n\}^2$ which is covered by infinite geodesics starting at the origin is at most an inverse power of $n$. This result is obtained without explicit limit shape assumptions.

math.PR↗
arXiv2021

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↗
arXiv2021

What does a typical metric space look like?

The collection $\mathcal{M}_n$ of all metric spaces on $n$ points whose diameter is at most $2$ can naturally be viewed as a compact convex subset of $\mathbb{R}^{\binom{n}{2}}$, known as the metric polytope. In this paper, we study the metric polytope for large $n$ and show that it is close to the cube $[1,2]^{\binom{n}{2}} \subseteq \mathcal{M}_n$ in the following two senses. First, the volume of the polytope is not much larger than that of the cube, with the following quantitative estimates: \[ \left(\tfrac{1}{6}+o(1)\right)n^{3/2} \le \log \mathrm{Vol}(\mathcal{M}_n)\le O(n^{3/2}). \] Second, when sampling a metric space from $\mathcal{M}_n$ uniformly at random, the minimum distance is at least $1 - n^{-c}$ with high probability, for some $c > 0$. Our proof is based on entropy techniques. We discuss alternative approaches to estimating the volume of $\mathcal{M}_n$ using exchangeability, Szemer\'edi's regularity lemma, the hypergraph container method, and the K\H{o}v\'ari--S\'os--Tur\'an theorem.

math.PR↗
arXiv2021

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 $\nabla \phi$ 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↗
arXiv2021

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↗
arXiv2020

Long-range order in discrete spin systems

We establish long-range order for discrete nearest-neighbor spin systems on $\mathbb{Z}^d$ satisfying a certain symmetry assumption, when the dimension $d$ is higher than an explicitly described threshold. The results characterize all periodic, maximal-pressure Gibbs states of the system. The results further apply in low dimensions provided that the lattice $\mathbb{Z}^d$ is replaced by $\mathbb{Z}^{d_1}\times\mathbb{T}^{d_2}$ with $d_1\ge 2$ and $d=d_1+d_2$ sufficiently high, where $\mathbb{T}$ is a cycle of even length. Applications to specific systems are discussed in detail and models for which new results are provided include the antiferromagnetic Potts model, Lipschitz height functions, and the hard-core, Widom--Rowlinson and beach models and their multi-type extensions. We also establish a formula conjectured by Jenssen and Keevash for the topological pressure in the high-dimensional limit.

math-ph↗
arXiv2020

Concentration inequalities for log-concave distributions with applications to random surface fluctuations

We derive two concentration inequalities for linear functions of log-concave distributions: an enhanced version of the classical Brascamp--Lieb concentration inequality, and an inequality quantifying log-concavity of marginals in a manner suitable for obtaining variance and tail probability bounds. These inequalities are applied to the statistical mechanics problem of estimating the fluctuations of random surfaces of the $\nabla\varphi$ type. The classical Brascamp--Lieb inequality bounds the fluctuations whenever the interaction potential is uniformly convex. We extend these bounds to the case of convex potentials whose second derivative vanishes only on a zero measure set, when the underlying graph is a $d$-dimensional discrete torus. The result applies, in particular, to potentials of the form $U(x)=|x|^p$ with $p>1$ and answers a question discussed by Brascamp--Lieb--Lebowitz (1975). Additionally, new tail probability bounds are obtained for the family of potentials $U(x) = |x|^p+x^2$, $p>2$. This result answers a question mentioned by Deuschel and Giacomin (2000).

math-ph↗
arXiv2020

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 $(n,x) \in [1,1+\delta) \times (1- \delta, 1]$ with $\delta >0$ 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 $n \in [1,1+\delta)$ and $x \in (1-\delta,1/\sqrt{n}]$ 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 $\omega$ is a collection of short loops and $\Gamma$ is a long loop, then the symmetric difference of $\omega$ and $\Gamma$ 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↗