Percolation on random recursive trees

We study Bernoulli bond percolation on a random recursive tree of size n with percolation parameter p(n) converging to 1 as n tends to infinity. The sizes of the percolation clusters are naturally stored in a tree structure. We prove convergence in distribution of this tree‐indexed process of cluster sizes to the genealogical tree of a continuous‐state branching process in discrete time. As a corollary we obtain the asymptotic sizes of the largest and next largest percolation clusters, extending thereby a recent work of Bertoin [5]. In a second part, we show that the same limit tree appears in the study of the tree components which emerge from a continuous‐time destruction of a random recursive tree. We comment on the connection to our first result on Bernoulli bond percolation. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 655–680, 2016     

Group-invariant Percolation on Graphs

Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, to properties of G-invariant percolation processes on X, such as the number of infinite components, the expected degree, and the topology of the components. Our fundamental tool is a new masstransport technique that has been occasionally used elsewhere and is developed further here.¶ Perhaps surprisingly, these investigations of group-invariant percolation produce results that are new in the Bernoulli setting. Most notably, we prove that critical Bernoulli percolation on any nonamenable Cayley graph has no infinite clusters. More generally, the same is true for any nonamenable graph with a unimodular transitive automorphism group.¶ We show that G is amenable if for all $ \alpha < 1 $ \alpha < 1 , there is a G-invariant site percolation process w \omega on X with $ {\bf P} [x \in \omega] > \alpha $ {\bf P} [x \in \omega] > \alpha for all vertices x and with no infinite components. When G is not amenable, a threshold $ \alpha < 1 $ \alpha < 1 appears. An inequality for the threshold in terms of the isoperimetric constant is obtained, extending an inequality of Häggström for regular trees.¶ If G acts transitively on X, we show that G is unimodular if the expected degree is at least 2 in any G-invariant bond percolation on X with all components infinite.¶ The investigation of dependent percolation also yields some results on automorphism groups of graphs that do not involve percolation.     

Supercritical loop percolation on Zd for d≥3

We consider supercritical percolation on ${\mathbb{Z}}^d$ ($d\ge 3$) induced by random walk loop soup. Two vertices are in the same cluster if they are connected through a sequence of intersecting loops. We obtain quenched parabolic Harnack inequalities, Gaussian heat kernel bounds, the invariance principle and the local central limit theorem for the simple random walks on the unique infinite cluster. We also show that the diameter of finite clusters have exponential tails like in Bernoulli bond percolation. Our results hold for all $d\ge 3$ and all supercritical intensities despite polynomial decay of correlations.     

Bond percolation critical probability bounds for three Archimedean lattices

Rigorous bounds for the bond percolation critical probability are determined for three Archimedean lattices: .7385 < p_c((3, 12²) bond) < .7449, .6430 < p_c((4, 6, 12) bond) < .7376, .6281 < p_c((4, 8²) bond) < .7201. Consequently, the bond percolation critical probability of the (3, 12²) lattice is strictly larger than those of the other ten Archimedean lattices. Thus, the (3, 12²) bond percolation critical probability is possibly the largest of any vertex‐transitive graph with bond percolation critical probability that is strictly less than one. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 20: 507–518, 2002     

Critical percolation: the expected number of clusters in a rectangle

We show that for critical site percolation on the triangular lattice two new observables have conformally invariant scaling limits. In particular the expected number of clusters separating two pairs of points converges to an explicit conformal invariant. Our proof is independent of earlier results and SLE techniques, and might provide a new approach to establishing conformal invariance of percolation.     

A Short Proof of the Harris-Kesten Theorem

We give a short proof of the fundamental result that the criticalprobability for bond percolation in the planar square latticeZ² is equal to 1/2. The lower bound was proved by Harris, whoshowed in 1960 that percolation does not occur at p = 1/2. Theother, more difficult, bound was proved by Kesten, who showedin 1980 that percolation does occur for any p > 1/2. 2000Mathematics Subject Classification 60K35, 82B43.     

Percolation in the hyperbolic plane

We study percolation in the hyperbolic plane and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such graphs, and Poisson-Voronoi-Bernoulli percolation. We prove the existence of three distinct nonempty phases for the Bernoulli processes. In the first phase, , there are no unbounded clusters, but there is a unique infinite cluster for the dual process. In the second phase, , there are infinitely many unbounded clusters for the process and for the dual process. In the third phase, , there is a unique unbounded cluster, and all the clusters of the dual process are bounded. We also study the dependence of in the Poisson-Voronoi-Bernoulli percolation process on the intensity of the underlying Poisson process.

     

Asymptotic of the Critical Value of the Large-Dimensional SIR Epidemic on Clusters

In this paper we are concerned with the susceptible-infective-removed (SIR) epidemic on open clusters of bond percolation on the square lattice. For the SIR model, a susceptible vertex is infected at rate proportional to the number of infective neighbors, while an infective vertex becomes removed at a constant rate. A removed vertex will never be infected again. We assume that at \(t=0\) the only infective vertex is the origin and define the critical value of the model as the supremum of the infection rates with which infective vertices die out with probability one; then, we show that the critical value under the annealed measure is \(\big (1+o(1)\big )/(2dp)\) as the dimension d of the lattice grows to infinity, where p is the probability that a given edge is open. Furthermore, we show that the critical value under the quenched measure equals the annealed one when the origin belongs to an infinite open cluster of the percolation.     

Two random walks on the open cluster of ℤ2 meet infinitely often

We prove that two independent continuous-time simple random walks on the infinite open cluster of a Bernoulli bond percolation in the lattice ?² meet each other infinitely many times. An application to the voter model is also discussed.     

Singularity of full scaling limits of planar nearcritical percolation

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.     

Quenched invariance principle for simple random walk on percolation clusters

We consider the simple random walk on the (unique) infinite cluster of super-critical bond percolation in ℤ ^d with d≥2. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square-integrable martingale. The size of the deformation, expressed by the so called corrector, is estimated by means of ergodicity arguments.     

Invariant percolation and harmonic Dirichlet functions

The main goal of this paper is to answer Question 1.10 and settle Conjecture 1.11 of Benjamini–Lyons–Schramm [BenLS] relating harmonic Dirichlet functions on a graph to those on the infinite clusters in the uniqueness phase of Bernoulli percolation. We extend the result to more general invariant percolations, including the random-cluster model. We prove the existence of the nonuniqueness phase for the Bernoulli percolation (and make some progress for random-cluster model) on unimodular transitive locally finite graphs admitting nonconstant harmonic Dirichlet functions. This is done by using the device of ℓ² Betti numbers. Received: May 2004 Revised: March 2005 Accepted: May 2005     

Percolation in a dependent random environment

Draw planes in ℝ³ that are orthogonal to the x axis, and intersect the x axis at the points of a Poisson process with intensity λ; similarly, draw planes orthogonal to the y and z axes using independent Poisson processes (with the same intensity). Taken together, these planes naturally define a randomly stretched rectangular lattice. Consider bond percolation on this lattice where each edge of length 𝓁 is open with probability e^−𝓁, and these events are independent given the edge lengths. We show that this model exhibits a phase transition: for large enough λ there is an infinite open cluster a.s., and for small λ all open clusters are finite a.s. We prove this result using the method of paths with exponential intersection tails, which is not applicable in two dimensions. The question whether the analogous process in the plane exhibits a phase transition is open. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 16, 333–343, 2000     

The critical probability for random Voronoi percolation in the plane is 1/2

We study percolation in the following random environment: let Z be a Poisson process of constant intensity on ℝ², and form the Voronoi tessellation of ℝ² with respect to Z. Colour each Voronoi cell black with probability p, independently of the other cells. We show that the critical probability is 1/2. More precisely, if p>1/2 then the union of the black cells contains an infinite component with probability 1, while if p<1/2 then the distribution of the size of the component of black cells containing a given point decays exponentially. These results are analogous to Kesten's results for bond percolation in ℤ². The result corresponding to Harris' Theorem for bond percolation in ℤ² is known: Zvavitch noted that one of the many proofs of this result can easily be adapted to the random Voronoi setting. For Kesten's results, none of the existing proofs seems to adapt. The methods used here also give a new and very simple proof of Kesten's Theorem for ℤ²; we hope they will be applicable in other contexts as well. Research supported in part by NSF grant ITR 0225610 and DARPA grant F33615-01-C-1900 Research partially undertaken during a visit to the Forschungsinstitut für Mathematik, ETH Zürich, Switzerland     

Monotonicity and phase diagram for multirange percolation on oriented trees

We consider Bernoulli bond percolation on oriented regular trees, where besides the usual short bonds, all bonds of a certain length are added. Independently, short bonds are open with probability p and long bonds are open with probability q. We study properties of the critical curve which delimits the set of pairs (p,q) for which there are almost surely no infinite paths. We also show that this curve decreases with respect to the length of the long bonds.     

Outlets of 2D invasion percolation and multiple-armed incipient infinite clusters

We study invasion percolation in two dimensions, focusing on properties of the outlets of the invasion and their relation to critical percolation and to incipient infinite clusters (IICs). First we compute the exact decay rate of the distribution of both the weight of the kth outlet and the volume of the kth pond. Next we prove bounds for all moments of the distribution of the number of outlets in an annulus. This result leads to almost sure bounds for the number of outlets in a box B(2 ⁿ ) and for the decay rate of the weight of the kth outlet to p _c . We then prove existence of multiple-armed IIC measures for any number of arms and for any color sequence which is alternating or monochromatic. We use these measures to study the invaded region near outlets and near edges in the invasion backbone far from the origin.     

Uniqueness of infinite rigid components in percolation models: the case of nonplanar lattices

We prove uniqueness of the infinite rigid component for standard bond percolation on periodic lattices in d-dimensional Euclidean space for arbitrary d, and more generally when the lattice is a quasi-transitive and amenable graph. Our approach to uniqueness of the infinite rigid component improves earlier ones, that were confined to planar settings.Research supported by the Swedish Research Council Mathematics Subject Classification (2000): 60K35, 82B43     

Brochette percolation

We study bond percolation on the square lattice with one-dimensional inhomogeneities. Inhomogeneities are introduced in the following way: A vertical column on the square lattice is the set of vertical edges that project to the same vertex on Z. Select vertical columns at random independently with a given positive probability. Keep (respectively remove) vertical edges in the selected columns, with probability p (respectively 1?p). All horizontal edges and vertical edges lying in unselected columns are kept (respectively removed) with probability q (respectively 1 ? q). We show that, if p > p_c(Z²) (the critical point for homogeneous Bernoulli bond percolation), then q can be taken strictly smaller than p_c(Z²) in such a way that the probability that the origin percolates is still positive.     

The Fourier spectrum of critical percolation

Consider the indicator function f of a 2-dimensional percolation crossing event. In this paper, the Fourier transform of f is studied and sharp bounds are obtained for its lower tail in several situations. Various applications of these bounds are derived. In particular, we show that the set of exceptional times of dynamical critical site percolation on the triangular grid in which the origin percolates has dimension \frac3136{\frac{31}{36}} almost surely, and the corresponding dimension in the half-plane is \frac59{\frac{5}{9}} . It is also proved that critical bond percolation on the square grid has exceptional times almost surely. Also, the asymptotics of the number of sites that need to be resampled in order to significantly perturb the global percolation configuration in a large square is determined.     

Infinite clusters and critical values in two-dimensional circle percolation

We consider a dependent percolation model onZ ² that does not have the ‘finite energy’ property. It is shown that the number of infinite clusters equals zero, one or infinity. Furthermore, we investigate a dynamical system which is associated with the calculation of the critical value in this model. It is shown that for almost all choices of the parameters in the model, this critical value can be calculated in a finite number of iterations.