We consider the (unoriented) long-range percolation on Z^d in dimensions d\ge1, where distinct sites x,y\in Z^d get connected with probability p_{xy}\in[0,1]. Assuming p_{xy}=|x-y|^{-s+o(1)} as |x-y|\to\infty, where s>0 and |\cdot| is a norm distance on Z^d, and supposing that the resulting random graph contains an infinite connected component C_{\infty}, we let D(x,y) be the graph distance between x and y measured on C_{\infty}. Our main result is that, for s\in(d,2d), D(x,y)=(\log|x-y|)^{Δ+o(1)},\qquad x,y\in C_{\infty}, |x-y|\to\infty, where Δ^{-1} is the binary logarithm of 2d/s and o(1) is a quantity tending to zero in probability as |x-y|\to\infty. Besides its interest for general percolation theory, this result sheds some light on a question that has recently surfaced in the context of ``small-world'' phenomena. As part of the proof we also establish tight bounds on the probability that the largest connected component in a finite box contains a positive fraction of all sites in the box.
The aim of this paper is to generalize the well-known asymptotic shape result for first-passage percolation on $\Zd$ to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet points to a deterministic shape that does not depend on the random environment. As a special case of the previous result, we obtain an asymptotic shape theorem for the chemical distance in supercritical Bernoulli percolation. We also prove a flat edge result. Some various examples are also given…
We continue our study of the chemical (graph) distance inside large critical percolation clusters in dimension two. We prove new estimates, which involve the three-arm probability, for the point-to-surface and point-to-point distances. We show that the point-to-point distance in $\mathbb{Z}^2$ between two points in the same critical percolation cluster has infinite second moment. We also give quantitative versions of our previous results comparing the length of the shortest crossing to that of the lowest crossing of a box.
We consider full scaling limits of planar nearcritical percolation in the Quad-Crossing-Topology introduced by Schramm and Smirnov. We show that two nearcritical scaling limits with different parameters are singular with respect to each other. The results hold for percolation models on rather general lattices, including bond percolation on the square lattice and site percolation on the triangular lattice.
We consider long-range percolation in dimension $d\geq 1$, where distinct sites $x$ and $y$ are connected with probability $p_{x,y}\in[0,1]$. Assuming that $p_{x,y}$ is translation invariant and that $p_{x,y}=\|x-y\|^{-s+o(1)}$ with $s>2d$, we show that the graph distance is at least linear with the Euclidean distance.
In this paper, we establish moderate deviations for the chemical distance in Bernoulli percolation. The chemical distance between two points is the length of the shortest open path between these two points. Thus, we study the size of random fluctuations around the mean value, and also the asymptotic behavior of this mean value. The estimates we obtain improve our knowledge of the convergence to the asymptotic shape. Our proofs rely on concentration inequalities proved by Boucheron, Lugosi and Massart, and also on the approximation theory of subadditive functions initiated by Alexander.
We initiate the study on chemical distances of percolation clusters for level sets of two-dimensional discrete Gaussian free fields as well as loop clusters generated by two-dimensional random walk loop soups. One of our results states that the chemical distance between two macroscopic annuli away from the boundary for the random walk loop soup at the critical intensity is of dimension 1 with positive probability. Our proof method is based on an interesting combination of a theorem of Makarov, isomorphism theory and an entropic repulsion estimate for Gaussian free fields in the presence of a h…
The ellipses model is a continuum percolation process in which ellipses with random orientation and eccentricity are placed in the plane according to a Poisson point process. A parameter $α$ controls the tail distribution of the major axis' distribution and we focus on the regime $α\in (1,2)$ for which there exists a unique infinite cluster of ellipses and this cluster fulfills the so called highway property. We prove that the distance within this infinite cluster behaves asymptotically like the (unrestricted) Euclidean distance in the plane. We also show that the chemical distance between poi…
We consider the percolation problem in the high-temperature Ising model on the two-dimensional square lattice at/near critical external fields. We show that all scaling relations, except a single hyperscaling relation, hold under the power law assumptions for the one-arm path and four-arm paths.
We consider two-dimensional critical bond percolation. Conditioned on the existence of an open circuit in an annulus, we show that the ratio of the expected size of the shortest open circuit to the expected size of the innermost circuit tends to zero as the side length of the annulus tends to infinity, the aspect ratio remaining fixed. The same proof yields a similar result for the lowest open crossing of a rectangle. In this last case, we answer a question of Kesten and Zhang by showing in addition that the ratio of the length of the shortest crossing to the length of the lowest tends to zero…
The chemical distance D(x,y) is the length of the shortest open path between two points x and y in an infinite Bernoulli percolation cluster. In this work, we study the asymptotic behaviour of this random metric, and we prove that, for an appropriate norm $μ$ depending on the dimension and the percolation parameter, the probability of the event \[\biggl\{0\leftrightarrow x,\frac{D(0,x)}{μ(x)}\notin (1-ε, 1+ε) \biggr\}\] exponentially decreases when $\|x\|_1$ tends to infinity. From this bound we also derive a large deviation inequality for the corresponding asymptotic shape result.
We provide the first nontrivial upper bound for the chemical distance exponent in two-dimensional critical percolation. Specifically, we prove that the expected length of the shortest horizontal crossing path of a box of side length $n$ in critical percolation on $\mathbb{Z}^2$ is bounded by $Cn^{2-δ}π_3(n)$, for some $δ>0$, where $π_3(n)$ is the "three-arm probability to distance $n$." This implies that the ratio of this length to the length of the lowest crossing is bounded by an inverse power of $n$ with high probability. In the case of site percolation on the triangular lattice, we obtain …
In this note, we describe some of the progress recently made on questions regarding the chemical distance in two-dimensional critical percolation by the author, J. Hanson, and P. Sosoe [6, 7]. It is expected that the distance between points in critical percolation clusters scales as $\|\cdot \|^{1+s}$, where $\|\cdot \|$ is the Euclidean distance and $s>0$. First, we review previous work of Aizenman-Burchard and Morrow-Zhang, which together establish a version of $0 < s \leq 1/3$. The main results of our work are in the direction of proving upper bounds on $s$, answering in [6] a question from…
In this paper we provide general conditions on a one parameter family of random infinite subsets of Z^d to contain a unique infinite connected component for which the chemical distances are comparable to the Euclidean distances, focusing primarily on models with long-range correlations. Our results are in the spirit of those by Antal and Pisztora proved for Bernoulli percolation. We also prove a shape theorem for balls in the chemical distance under such conditions. Our general statements give novel results about the structure of the infinite connected component of the vacant set of random int…
We prove that, after centering and diffusively rescaling space and time, the collection of rightmost infinite open paths in a supercritical oriented percolation configuration on the space-time lattice Z^2_{even}:={(x,i) in Z^2: x+i is even} converges in distribution to the Brownian web. This proves a conjecture of Wu and Zhang. Our key observation is that each rightmost infinite open path can be approximated by a percolation exploration cluster, and different exploration clusters evolve independently before they intersect.
The understanding of site percolation on the triangular lattice progressed greatly in the last decade. Smirnov proved conformal invariance of critical percolation, thus paving the way for the construction of its scaling limit. Recently, the scaling limit of near-critical percolation was also constructed by Garban, Pete and Schramm. The aim of this very modest contribution is to explain how these results imply the convergence, as p tends to p_c, of the Wulff crystal to a Euclidean disk. The main ingredient of the proof is the rotational invariance of the scaling limit of near-critical percolati…
We consider an i.i.d. supercritical bond percolation on Z^d , every edge is open with a probability p > p\_c (d), where p\_c (d) denotes the critical parameter for this percolation. We know that there exists almost surely a unique infinite open cluster C\_p [11]. We are interested in the regularity properties of the chemical distance for supercritical Bernoulli percolation. The chemical distance between two points x, y $\in$ C\_p corresponds to the length of the shortest path in C\_p joining the two points. The chemical distance between 0 and nx grows asymptotically like n$μ$\_p (x). We aim to…
Under some general assumptions, we construct the scaling limit of open clusters and their associated counting measures in a class of two dimensional percolation models. Our results apply, in particular, to critical Bernoulli site percolation on the triangular lattice and to the critical FK-Ising model on the square lattice. Fundamental properties of the scaling limit, such as conformal covariance, are explored. As an application to Bernoulli percolation, we obtain the scaling limit of the largest cluster in a bounded domain. We also apply our results to the critical, two-dimensional Ising mode…
We show that the laws of scaling limits of nearcritical percolation exploration paths with different parameters are singular with respect to each other. This generalises a result of Nolin and Werner, using a similar technique. As a corollary, the singularity can even be detected from an infinitesimal initial segment. Moreover, nearcritical scaling limits of exploration paths are mutually singular under scaling maps.
We prove existence of the scaling limit of the invasion percolation cluster (IPC) on a regular tree. The limit is a random real tree with a single end. The contour and height functions of the limit are described as certain diffusive stochastic processes. This convergence allows us to recover and make precise certain asymptotic results for the IPC. In particular, we relate the limit of the rescaled level sets of the IPC to the local time of the scaled height function.
We extend Smirnov's proof of the existence and conformal invariance of the scaling limit of critical site-percolation on the triangular lattice to particular sequences of periodic graphs with more arbitrary large-scale structure, obtained by piecing together triangular regions of the triangular lattice. While not formally speaking a scaling limit statement (as the graphs are not rescaled versions of each other), the result is a weak form of universality for critical percolation.
We derive an estimate for the distance, measured in lattice spacings, inside two-dimensional critical percolation clusters from the origin to the boundary of the box of side length $2n$, conditioned on the existence of an open connection. The estimate we obtain is the radial analogue of the one found in the work of Damron, Hanson, and Sosoe. In the present case, however, there is no lowest crossing in the box to compare to, so we construct a path $γ$ from the origin to distance $n$ that consists of "three-arm" points, and whose volume can thus be estimated by $O(n^2π_3(n))$. Here, $π_3(n)$ is …
Let $\{Y_{\mathfrak{B}}(x)\,:\,x\in\mathfrak{B}\}$ be a discrete Gaussian free field in a two-dimensional box $\mathfrak{B}$ of side length $S$ with Dirichlet boundary conditions. We study Liouville first-passage percolation: the shortest-path metric in which each vertex $x$ is given a weight of $e^{γY_{\mathfrak{B}}(x)}$ for some $γ>0$. We show that for sufficiently small but fixed $γ>0$, for any sequence of scales $\{S_{k}\}$ there exists a subsequence along which the appropriately scaled and interpolated Liouville FPP metric converges in the Gromov--Hausdorff sense to a random metric on the…
We study the possible scaling limits of percolation interfaces in two dimensions on the triangular lattice. When one lets the percolation parameter p(N) vary with the size N of the box that one is considering, three possibilities arise in the large-scale limit. It is known that when p(N) does not converge to 1/2 fast enough, then the scaling limits are degenerate, whereas if p(N) - 1 / 2 goes to zero quickly, the scaling limits are SLE(6) as when p=1/2. We study some properties of the (non-void) intermediate regime where the large scale behavior is neither SLE(6) nor degenerate. We prove that …
Jigsaw percolation is a nonlocal process that iteratively merges connected clusters in a deterministic "puzzle graph" by using connectivity properties of a random "people graph" on the same set of vertices. We presume the Erdos--Renyi people graph with edge probability p and investigate the probability that the puzzle is solved, that is, that the process eventually produces a single cluster. In some generality, for puzzle graphs with N vertices of degrees about D (in the appropriate sense), this probability is close to 1 or small depending on whether pD(log N) is large or small. The one dimens…
We study gradient percolation for site percolation on the triangular lattice. This is a percolation model where the percolation probability depends linearly on the location of the site. We prove the results predicted by physicists for this model. More precisely, we describe the fluctuations of the interfaces around their (straight) scaling limits, and the expected and typical lengths of these interfaces. These results build on the recent results for critical percolation on this lattice by Smirnov, Lawler, Schramm and Werner, and on the scaling ideas developed by Kesten.
In exponential last passage percolation, we consider the rescaled Busemann process $x\mapsto N^{-1/3}B^ρ_{0,[xN^{2/3}]e_1} \,\, (x\in\mathbb{R})$, as a process parametrized by the scaled density $ρ=1/2+\fracμ{4} N^{-1/3}$, and taking values in $C(\mathbb{R})$. We show that these processes, as $N\rightarrow \infty$, have a càdlàg scaling limit $G=(G_μ)_{μ\in \mathbb{R}}$, parametrized by $μ$ and taking values in $C(\mathbb{R})$. The limiting process $G$, which can be thought of as the Busemann process under the KPZ scaling, can be described as an ensemble of "sticky" lines of Brownian regularit…
In a recent breakthrough work, Chatterjee [Ann. of Math. (2) 177 (2013) 663-697] proved a long standing conjecture that relates the transversal exponent $ξ$ and the fluctuation exponent $χ$ in first-passage percolation on $\mathbb{Z}^d$. The purpose of this paper is to replace the main argument of Chatterjee (2013) and give an alternative proof of this relation. Specifically, we show that under the assumption that exponents defined in Chatterjee (2013) exist, one has the relation $χ\leq2ξ-1$. One advantage of our argument is that it does not require the "nearly Gamma" assumption of Chatterjee …
We study first passage percolation (FPP) with stationary edge weights on Cayley graphs of finitely generated virtually nilpotent groups. Previous works of Benjamini-Tessera and Cantrell-Furman show that scaling limits of such FPP are given by Carnot-Carathéodory metrics on the associated graded nilpotent Lie group. We show a converse, i.e. that for any Cayley graph of a finitely generated nilpotent group, any Carnot-Carathéodory metric on the associated graded nilpotent Lie group is the scaling limit of some FPP with stationary edge weights on that graph. Moreover, for any Cayley graph of any …
We prove Tsirelson's conjecture that any scaling limit of the critical planar percolation is a black noise. Our theorems apply to a number of percolation models, including site percolation on the triangular grid and any subsequential scaling limit of bond percolation on the square grid. We also suggest a natural construction for the scaling limit of planar percolation, and more generally of any discrete planar model describing connectivity properties.
We prove that near-critical percolation and dynamical percolation on the triangular lattice $η\mathbb{T}$ have a scaling limit as the mesh $η\to 0$, in the "quad-crossing" space $\mathcal{H}$ of percolation configurations introduced by Schramm and Smirnov. The proof essentially proceeds by "perturbing" the scaling limit of the critical model, using the pivotal measures studied in our earlier paper. Markovianity and conformal covariance of these new limiting objects are also established.
In bootstrap percolation it is known that the critical percolation threshold tends to converge slowly to zero with increasing system size, or, inversely, the critical size diverges fast when the percolation probability goes to zero. To obtain higher-order terms (that is, sharp and sharper thresholds) for the percolation threshold in general is a hard question. In the case of two-dimensional anisotropic models, sometimes correction terms can be obtained from inversion in a relatively simple manner.
The scaling limit of the critical percolation, is it a black noise? The answer depends on stability to perturbations concentrated along a line. This text, containing no proofs, reports experimental results that suggest the affirmative answer.
We study families of dependent site percolation models on the triangular lattice ${\mathbb T}$ and hexagonal lattice ${\mathbb H}$ that arise by applying certain cellular automata to independent percolation configurations. We analyze the scaling limit of such models and show that the distance between macroscopic portions of cluster boundaries of any two percolation models within one of our families goes to zero almost surely in the scaling limit. It follows that each of these cellular automaton generated dependent percolation models has the same scaling limit (in the sense of Aizenman-Burchard…
We show quenched large deviations for the simple random walk on a certain class of percolations with long-range correlations. This class contains the supercritical Bernoulli percolations, the model considered by Drewitz, R'ath and Sapozhnikov and the random-cluster model up to the slab critical point. Our result is an extension of Kubota's result for the supercritical Bernoulli percolations. We also state a shape theorem for the chemical distance, which is an extension of Garet and Marchand's result for the supercritical Bernoulli percolations.
We prove annealed scaling relations for planar Voronoi percolation. To our knowledge, this is the first result of this kind for a continuum percolation model. We are mostly inspired by the proof of scaling relations for Bernoulli percolation by Kesten [Kes87]. Along the way, we show an annealed quasi-multiplicativity property by relying on the quenched box-crossing property proved by Ahlberg, Griffiths, Morris and Tassion [AGMT16]. Intermediate results also include the study of quenched and annealed notions of pivotal events and the extension of the quenched box-crossing property of [AGMT16] t…
In this article, we consider an anisotropic finite-range bond percolation model on $\mathbb{Z}^2$. On each horizontal layer $\{(x,i): x \in \mathbb{Z}\}$ we have edges $\langle(x, i),(y, i)\rangle$ for $1 \leq |x - y| \leq N$. There are also vertical edges connecting two nearest neighbor vertices on distinct layers $\langle(x, i), (x, i+1)\rangle$ for $x, i \in\mathbb{Z}$. On this graph we consider the following anisotropic independent percolation model: horizontal edges are open with probability $1/(2N)$, while vertical edges are open with probability $ε$ to be suitably tuned as $N$ grows to …
We present a review of the recent progress on percolation scaling limits in two dimensions. In particular, we will consider the convergence of critical crossing probabilities to Cardy's formula and of the critical exploration path to chordal SLE(6), the full scaling limit of critical cluster boundaries, and near-critical scaling limits.
We consider several aspects of the scaling limit of percolation on random planar triangulations, both finite and infinite. The equivalents for random maps of Cardy's formula for the limit under scaling of various crossing probabilities are given. The limit probabilities are expressed in terms of simple events regarding Airy-Levy processes. Some explicit formulas for limit probabilities follow from this relation by applying known results on stable processes. Conversely, natural symmetries of the random maps imply identities concerning the Airy-Levy processes.
We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on a version of the triangular lattice in the complex plane have unique scaling limits, which are invariant under rotations, scalings, and, in the case of the MST, also under translations. However, they are not expected to be conformally invariant. We also prove some geometric properties of the limiting MST. The topology of convergence is the space of spanning trees introduced by Aizenman, Burchard, Newman & Wilson (1999), and the proof relies on the existence and conformal covariance of the scaling limit of the near-cri…
We show that random walk on the incipient infinite cluster (IIC) of two-dimensional critical percolation is subdiffusive in the chemical distance (i.e., in the intrinsic graph metric). Kesten (1986) famously showed that this is true for the Euclidean distance, but it is known that the chemical distance is typically asymptotically larger. More generally, we show that subdiffusivity in the chemical distance holds for stationary random graphs of polynomial volume growth, as long as there is a multi-scale way of covering the graph so that "deep patches" have "thin backbones". Our estimates are qua…
We provide a new proof of the near-critical scaling relation $β=ξ_1ν$ for Bernoulli percolation on the square lattice already proved by Kesten in 1987. We rely on a novel approach that does not invoke Russo's formula, but rather relates differences in crossing probabilities at different scales. The argument is shorter and more robust than previous ones and is more likely to be adapted to other models. The same approach may be used to prove the other scaling relations appearing in Kesten's work.
We use SLE(6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice -- that is, the scaling limit of the set of all interfaces between different clusters. Some properties of the loop process, including conformal invariance, are also proved.
We analyze the geometry of scaling limits of near-critical 2D percolation, i.e., for $p=p_c+λδ^{1/ν}$, with $ν=4/3$, as the lattice spacing $δ\to 0$. Our proposed framework extends previous analyses for $p=p_c$, based on $SLE_6$. It combines the continuum nonsimple loop process describing the full scaling limit at criticality with a Poissonian process for marking double (touching) points of that (critical) loop process. The double points are exactly the continuum limits of "macroscopically pivotal" lattice sites and the marked ones are those that actually change state as $λ$ varies. This struc…
This is an introductory account of the emergence of conformal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov's theorem (2001) on the conformal invariance of crossing probabilities in site percolation on the triangular lattice. We also give an introductory account of Schramm-Loewner evolutions (SLE(k)), a one-parameter family of conformally invariant random curves discovered by Schramm (2000). The article is organized around the aim of proving the result, due to Smirnov (2001) and to Camia and Newman (2007), that the percolation exploration path…
It has been conjectured in numerous physics papers that in ordinary first-passage percolation on integer lattices, the fluctuation exponent $χ$ and the wandering exponent $ξ$ are related through the universal relation $χ=2ξ-1$, irrespective of the dimension. This is sometimes called the KPZ relation between the two exponents. This article gives a rigorous proof of this conjecture assuming that the exponents exist in a certain sense.
Consider a cellular automaton with state space $\{0,1 \}^{{\mathbb Z}^2}$ where the initial configuration $ω_0$ is chosen according to a Bernoulli product measure, 1's are stable, and 0's become 1's if they are surrounded by at least three neighboring 1's. In this paper we show that the configuration $ω_n$ at time n converges exponentially fast to a final configuration $\barω$, and that the limiting measure corresponding to $\barω$ is in the universality class of Bernoulli (independent) percolation. More precisely, assuming the existence of the critical exponents $β$, $η$, $ν$ and $γ$, and of …
In the mean field (or random link) model there are $n$ points and inter-point distances are independent random variables. For $0 < \ell < \infty$ and in the $n \to \infty$ limit, let $δ(\ell) = 1/n \times$ (maximum number of steps in a path whose average step-length is $\leq \ell$). The function $δ(\ell)$ is analogous to the percolation function in percolation theory: there is a critical value $\ell_* = e^{-1}$ at which $δ(\cdot)$ becomes non-zero, and (presumably) a scaling exponent $β$ in the sense $δ(\ell) \asymp (\ell - \ell_*)^β$. Recently developed probabilistic methodology (in some sens…
We study scaling limits and conformal invariance of critical site percolation on triangular lattice. We show that some percolation-related quantities are harmonic conformal invariants, and calculate their values in the scaling limit. As a particular case we obtain conformal invariance of the crossing probabilities and Cardy's formula. Then we prove existence, uniqueness, and conformal invariance of the continuum scaling limit.
