Supercritical Percolation Research Articles

Supercritical percolation on finite transitive graphs I: Uniqueness of the giant component

Supercritical percolation on finite transitive graphs I: Uniqueness of the giant component

Barely Supercritical Percolation on Poissonian Scale-free Networks

We study the giant component problem slightly above the critical regime for percolation on Poissonian random graphs in the scale-free regime, where the vertex weights and degrees have a diverging second moment. Critical percolation on scale-free random graphs has been observed to have incredibly subtle features that are markedly different compared to those in random graphs with a converging second moment. In particular, the critical window for percolation depends sensitively on whether we consider single- or multi-edge versions of the Poissonian random graph. In this paper, and together with our companion paper [3], we build a bridge between these two cases. Our results characterize the part of the barely supercritical regime where the size of the giant components are approximately same for the single- and multi-edge settings. The methods for establishing concentration of giant for the single- and multi-edge versions are quite different. While the analysis in the multi-edge case is based on scaling limits of exploration processes, the single-edge setting requires identi fication of a core structure inside certain high-degree vertices that forms the giant component.

Isoperimetric Inequalities and Supercritical Percolation on High-Dimensional Graphs

It is known that many different types of finite random subgraph models undergo quantitatively similar phase transitions around their percolation thresholds, and the proofs of these results rely on isoperimetric properties of the underlying host graph. Recently, the authors showed that such a phase transition occurs in a large class of regular high-dimensional product graphs, generalising a classic result for the hypercube. In this paper we give new isoperimetric inequalities for such regular high-dimensional product graphs, which generalise the well-known isoperimetric inequality of Harper for the hypercube, and are asymptotically sharp for a wide range of set sizes. We then use these isoperimetric properties to investigate the structure of the giant component L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} in supercritical percolation on these product graphs, that is, when p=1+ϵd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=\\frac{1+\\epsilon }{d}$$\\end{document}, where d is the degree of the product graph and ϵ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon >0$$\\end{document} is a small enough constant. We show that typically L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} has edge-expansion Ω1dlnd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\left( \\frac{1}{d\\ln d}\\right) $$\\end{document}. Furthermore, we show that L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} likely contains a linear-sized subgraph with vertex-expansion Ω1dlnd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\left( \\frac{1}{d\\ln d}\\right) $$\\end{document}. These results are best possible up to the logarithmic factor in d. Using these likely expansion properties, we determine, up to small polylogarithmic factors in d, the likely diameter of L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} as well as the typical mixing time of a lazy random walk on L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document}. Furthermore, we show the likely existence of a cycle of length Ωndlnd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\left( \\frac{n}{d\\ln d}\\right) $$\\end{document}. These results not only generalise, but also improve substantially upon the known bounds in the case of the hypercube, where in particular the likely diameter and typical mixing time of L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L_1$$\\end{document} were previously only known to be polynomial in d.

Supercritical percolation on graphs of polynomial growth

Supercritical percolation on graphs of polynomial growth

Slightly supercritical percolation on nonamenable graphs II: growth and isoperimetry of infinite clusters

Slightly supercritical percolation on nonamenable graphs II: growth and isoperimetry of infinite clusters

Transience and anchored isoperimetric dimension of supercritical percolation clusters

We establish several equivalent characterisations of the anchored isoperimetric dimension of supercritical clusters in Bernoulli bond percolation on transitive graphs. We deduce from these characterisations together with a theorem of Duminil-Copin, Goswami, Raoufi, Severo, and Yadin (Duke Math. J. 2020) that if G is a transient transitive graph then the infinite clusters of Bernoulli percolation on G are transient for p sufficiently close to 1. It remains open to extend this result down to the critical probability. Along the way we establish two new cluster repulsion inequalities that are of independent interest.

Hydrodynamic limit of simple exclusion processes in symmetric random environments via duality and homogenization

We consider continuous-time random walks on a random locally finite subset of mathbb {R}^d with random symmetric jump probability rates. The jump range can be unbounded. We assume some second-moment conditions and that the above randomness is left invariant by the action of the group mathbb {G}=mathbb {R}^d or mathbb {G}=mathbb {Z}^d. We then add a site-exclusion interaction, thus making the particle system a simple exclusion process. We show that, for almost all environments, under diffusive space–time rescaling the system exhibits a hydrodynamic limit in path space. The hydrodynamic equation is non-random and governed by the effective homogenized matrix D of the single random walk, which can be degenerate. The above result covers a very large family of models including e.g. simple exclusion processes built from random conductance models on mathbb {Z}^d and on crystal lattices (possibly with long conductances), Mott variable range hopping, simple random walks on Delaunay triangulations, random walks on supercritical percolation clusters.

Slightly supercritical percolation on non‐amenable graphs I: The distribution of finite clusters

We study the distribution of finite clusters in slightly supercritical ( p ↓ p c $p \downarrow p_c$ ) Bernoulli bond percolation on transitive non-amenable graphs, proving in particular that if G $G$ is a transitive non-amenable graph satisfying the L 2 $L^2$ boundedness condition ( p c < p 2 → 2 $p_c<p_{2\rightarrow 2}$ ) and K $K$ denotes the cluster of the origin and then there exists δ > 0 $\delta >0$ such that if p ∈ ( p c − δ , p c + δ ) $p\in (p_c-\delta ,p_c+\delta )$ , then P p ( n ⩽ | K | < ∞ ) ≍ n − 1 / 2 exp − Θ | p − p c | 2 n \begin{align*} \hskip1.5pc\mathbf {P}_p(n \leqslant |K| < \infty ) &\asymp n^{-1/2} \,{\exp} {\left[ -\Theta {\left(|p-p_c|^2 n\right)} \right]}\hskip-1.5pc \end{align*} and P p ( r ⩽ Rad ( K ) < ∞ ) ≍ r − 1 exp [ − Θ ( | p − p c | r ) ] \begin{align*} \hskip1.5pc\mathbf {P}_p(r \leqslant \operatorname{Rad}(K) < \infty ) &\asymp r^{-1} \exp {[ -\Theta (|p-p_c| r) ]}\hskip-1.5pc \end{align*} for every n , r ⩾ 1 $n,r\geqslant 1$ , where all implicit constants depend only on G $G$ . We deduce in particular that the critical exponents γ ′ $\gamma ^{\prime }$ and Δ ′ $\Delta ^{\prime }$ describing the rate of growth of the moments of a finite cluster as p ↓ p c $p \downarrow p_c$ take their mean-field values of 1 and 2, respectively. These results apply in particular to Cayley graphs of non-elementary hyperbolic groups, to products with trees, and to transitive graphs of spectral radius ρ < 1 / 2 $\rho <1/2$ . In particular, every finitely generated non-amenable group has a Cayley graph to which these results apply. They are new for graphs that are not trees. The corresponding facts are yet to be understood on Z d $\mathbb {Z}^d$ even for d $d$ very large. In a second paper in this series, we will apply these results to study the geometric and spectral properties of infinite slightly supercritical clusters in the same setting.

Biased random walk on supercritical percolation: anomalous fluctuations in the ballistic regime

We study biased random walk on the infinite connected component of supercritical percolation on the integer lattice Zd for d≥2. For this model, Fribergh and Hammond showed the existence of an exponent γ such that: for γ<1, the random walk is sub-ballistic (i.e. has zero velocity asymptotically), with polynomial escape rate described by γ; whereas for γ>1, the random walk is ballistic, with non-zero speed in the direction of the bias. They moreover established, under the usual diffusive scaling about the mean distance travelled by the random walk in the direction of the bias, a central limit theorem when γ>2. In this article, we explain how Fribergh and Hammond’s percolation estimates further allow it to be established that for γ∈(1,2) the fluctuations about the mean are of an anomalous polynomial order, with exponent given by γ−1.

The asymptotic distribution of cluster sizes for supercritical percolation on random split trees

AbstractWe consider the model of random trees introduced by Devroye, the so‐called random split trees. The model encompasses many important randomized algorithms and data structures. We then perform supercritical Bernoulli bond‐percolation on those trees and obtain a precise weak limit theorem for the sizes of the largest clusters. We also show that the approach developed in this work may be useful for studying percolation on other classes of trees with logarithmic height, for instance, we also study the case of d‐regular trees.

Continuum percolation and stochastic epidemic models on Poisson and Ginibre point processes

The most studied continuum percolation model in two dimensions is the Boolean model consisting of disks with the same radius whose centers are randomly distributed on the Poisson point process (PPP). We also consider the Boolean percolation model on the Ginibre point process (GPP) which is a typical repelling point process realizing hyperuniformity. We think that the PPP approximates a disordered configuration of individuals, while the GPP does a configuration of citizens adopting a strategy to keep social distancing in a city in order to avoid contagion. We consider the SIR models with contagious infection on supercritical percolation clusters formed on the PPP and the GPP. By numerical simulations, we studied dependence of the percolation phenomena and the infection processes on the PPP- and the GPP-underlying graphs. We show that in a subcritical regime of infection rate the PPP-based models show emergence of infection clusters on clumping of points which is formed by fluctuation of uncorrelated Poissonian statistics. On the other hand, the cumulative numbers of infected individuals in processes are suppressed in the GPP-based models.

Duality of random planar maps via percolation

We discuss duality properties of critical Boltzmann planar maps such that the degree of a typical face is in the domain of attraction of a stable distribution with parameter $\alpha\in(1,2]$. We consider the critical Bernoulli bond percolation model on a Boltzmann map in the dilute and generic regimes $\alpha \in (3/2,2]$, and show that the open percolation cluster of the origin is itself a Boltzmann map in the dense regime $\alpha \in (1,3/2)$, with parameter \[\alpha':= \frac{2\alpha+3}{4\alpha-2}.\] This is the counterpart in random planar maps of the duality property $\kappa \leftrightarrow 16/\kappa$ of Schramm--Loewner Evolutions and Conformal Loop Ensembles, recently established by Miller, Sheffield and Werner. As a byproduct, we identify the scaling limit of the boundary of the percolation cluster conditioned to have a large perimeter. The cases of subcritical and supercritical percolation are also discussed. In particular, we establish the sharpness of the phase transition through the tail distribution of the size of the percolation cluster.

Quantitative homogenization of the parabolic and elliptic Green’s functions on percolation clusters

We study the heat kernel and the Green’s function on the infinite supercritical percolation cluster in dimension d≥2 and prove a quantitative homogenization theorem for these functions with an almost optimal rate of convergence. These results are a quantitative version of the local central limit theorem proved by Barlow and Hambly in (Electron. J. Probab. 14 (2009) 1–27). The proof relies on a structure of renormalization for the infinite percolation cluster introduced in (Comm. Pure Appl. Math. 71 (2018) 1717–1849), Gaussian bounds on the heat kernel established by Barlow in (Ann. Probab. 32 (2004) 3024–3084) and tools of the theory of quantitative stochastic homogenization. An important step in the proof is to establish a C0,1-large-scale regularity theory for caloric functions on the infinite cluster and is of independent interest.

Optimal corrector estimates on percolation cluster

We prove optimal quantitative estimates on the first-order correctors on supercritical percolation clusters: we show that they are bounded in dimension larger than 3 and have logarithmic growth in dimension 2 in the sense of stretched exponential moments. The main ingredients are a renormalization scheme of the supercritical percolation cluster, following the works of Pisztora (Probab. Theory Related Fields 104 (1996) 427–466); large-scale regularity estimates developed by Armstrong and the author in (Comm. Pure Appl. Math. 71 (2018) 1717–1849); and a nonlinear concentration inequality of the Efron–Stein type which is used to transfer quantitative information from the environment to the correctors.

Supercritical percolation on nonamenable graphs: isoperimetry, analyticity, and exponential decay of the cluster size distribution

Let G be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on G. We prove that if G is nonamenable and p > p_c(G) then there exists a positive constant c_p such thatPp(n≤|K|<∞)≤e-cpn\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\mathbf {P}_p(n \\le |K| < \\infty ) \\le e^{-c_p n} \\end{aligned}$$\\end{document}for every nge 1, where K is the cluster of the origin. We deduce the following two corollaries:Every infinite cluster in supercritical percolation on a transitive nonamenable graph has anchored expansion almost surely. This answers positively a question of Benjamini et al. (in: Random walks and discrete potential theory (Cortona, 1997), symposium on mathematics, XXXIX, Cambridge University Press, Cambridge, pp 56–84, 1999).For transitive nonamenable graphs, various observables including the percolation probability, the truncated susceptibility, and the truncated two-point function are analytic functions of p throughout the supercritical phase.

The exclusion process mixes (almost) faster than independent particles

Oliveira conjectured that the order of the mixing time of the exclusion process with $k$-particles on an arbitrary $n$-vertex graph is at most that of the mixing-time of $k$ independent particles. We verify this up to a constant factor for $d$-regular graphs when each edge rings at rate $1/d$ in various cases: (1) when $d=\Omega (\log _{n/k}n)$, (2) when $\mathrm{gap}:=$ the spectral-gap of a single walk is $O(1/\log ^{4}n)$ and $k\ge n^{\Omega (1)}$, (3) when $k\asymp n^{a}$ for some constant $0<a<1$. In these cases, our analysis yields a probabilistic proof of a weaker version of Aldous’ famous spectral-gap conjecture (resolved by Caputo et al.). We also prove a general bound of $O(\log n\log \log n/\mathrm{gap})$, which is within a $\log \log n$ factor from Oliveira’s conjecture when $k\ge n^{\Omega (1)}$. As applications, we get new mixing bounds: (a) $O(\log n\log \log n)$ for expanders, (b) order $d\log (dk)$ for the hypercube $\{0,1\}^{d}$, (c) order $(\mathrm{Diameter})^{2}\log k$ for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus.

Random conductance models with stable-like jumps: Heat kernel estimates and Harnack inequalities

We establish two-sided heat kernel estimates for random conductance models with non-uniformly elliptic (possibly degenerate) stable-like jumps on graphs. These are long range counterparts of the well known two-sided Gaussian heat kernel estimates by M.T. Barlow for nearest neighbor (short range) random walks on the supercritical percolation cluster. Unlike the cases of nearest neighbor conductance models, we cannot use parabolic Harnack inequalities since even elliptic Harnack inequalities do not hold in the present setting. As an application, we establish the local limit theorem for the models.

Anchored isoperimetric profile of the infinite cluster in supercritical bond percolation is Lipschitz continuous

We consider the standard model of i.i.d. first passage percolation on $\mathbb {Z}^{d}$ given a distribution $G$ on $\mathbb {R}_{+}$. We consider a cube oriented in the direction $\overrightarrow {v}$ whose sides have length $n$. We study the maximal flow from the top half to the bottom half of the boundary of this cube. We already know that the maximal flow renormalized by $n^{d-1}$ converges towards the flow constant $\nu _{G}(\overrightarrow {v})$. We prove here that the map $p\mapsto \nu _{p\delta _{1}+(1-p)\delta _{0}}$ is Lipschitz continuous on all intervals $[p_{0},p_{1}]\subset (p_{c}(d),1)$ where $p_{c}(d)$ denotes the critical parameter for i.i.d. bond percolation on $\mathbb {Z}^{d}$. For $p>p_{c}(d)$, we know that there exists almost surely a unique infinite open cluster $\mathcal {C}_{p}$ [8]. We are interested in the regularity properties in $p$ of the anchored isoperimetric profile of the infinite cluster $\mathcal {C}_{p}$. For $d\geq 2$, using the result on the regularity of the flow constant, we prove here that the anchored isoperimetric profile defined in [4] is Lipschitz continuous on all intervals $[p_{0},p_{1}]\subset (p_{c}(d),1)$.

Green kernel asymptotics for two-dimensional random walks under random conductances

We consider random walks among random conductances on $\mathbb{Z}^2$ and establish precise asymptotics for the associated potential kernel and the Green's function of the walk killed upon exiting balls. The result is proven for random walks on i.i.d. supercritical percolation clusters among ergodic degenerate conductances satisfying a moment condition. We also provide a similar result for the time-dynamic random conductance model. As an application we present a scaling limit for the variances in the Ginzburg-Landau $\nabla \phi$-interface model.

Continuity for the Rate Function of the Simple Random Walk on Supercritical Percolation Clusters

We consider the simple random walk on supercritical percolation clusters in the multidimensional cubic lattice. In this model, a quenched large deviation principle holds for the position of the random walk. Its rate function depends on the law of the percolation configuration, and the aim of this paper is to study the continuity of the rate function in the law. To do this, it is useful that the rate function is expressed by the so-called Lyapunov exponent, which is the asymptotic cost paid by the random walk for traveling in a landscape of percolation configurations. In this context, we first observe the continuity of the Lyapunov exponent in the law of the percolation configuration and then lift it to the rate function.