Supercritical Percolation Research Articles

Transience of percolation clusters on wedges

We study random walks on supercritical percolation clusters on wedges in $Z^3$, and show that the infinite percolation cluster is (a.s.) transient whenever the wedge is transient. This solves a question raised by O. Häggström and E. Mossel. We also show that for convex gauge functions satisfying a mild regularity condition, the existence of a finite energy flow on $Z^2$ is equivalent to the (a.s.) existence of a finite energy flow on the supercritical percolation cluster. This answers a question of C. Hoffman.

Random walks on supercritical percolation clusters

We obtain Gaussian upper and lower bounds on the transition density qt(x;y) of the continuous time simple random walk on a supercritical percolation cluster C1 in the Euclidean lattice. The bounds, analogous to Aronsen's bounds for uniformly elliptic divergence form diusions, hold with constants ci depending only on p (the percolation probability) and d. The irregular nature of the medium means that the bound for qt(x; ) only holds for t Sx(!), where the constant Sx(!) depends on the percolation congura- tion !.

Anchored expansion, percolation and speed

Benjamini, Lyons and Schramm [Random Walks and Discrete Potential Theory (1999) 56–84] considered properties of an infinite graph G, and the simple random walk on it, that are preserved by random perturbations. In this paper we solve several problems raised by those authors. The anchored expansion constant is a variant of the Cheeger constant; its positivity implies positive lower speed for the simple random walk, as shown by Virág [Geom. Funct. Anal. 10 (2000) 1588–1605]. We prove that if G has a positive anchored expansion constant, then so does every infinite cluster of independent percolation with parameter p sufficiently close to 1; a better estimate for the parameters p where this holds is in the Appendix. We also show that positivity of the anchored expansion constant is preserved under a random stretch if and only if the stretching law has an exponential tail. We then study a simple random walk in the infinite percolation cluster in Cayley graphs of certain amenable groups known as “lamplighter groups.” We prove that zero speed for a random walk on a lamplighter group implies zero speed for random walk on an infinite cluster, for any supercritical percolation parameter p. For p large enough, we also establish the converse.

The speed of biased random walk on percolation clusters

We consider biased random walk on supercritical percolation clusters in ℤ2. We show that the random walk is transient and that there are two speed regimes: If the bias is large enough, the random walk has speed zero, while if the bias is small enough, the speed of the random walk is positive.

The Speed of Simple Random Walk and Anchored Expansion on Percolation Clusters: an Overview

Benjamini, Lyons and Schramm (1999) considered properties of an infinite graph $G$, and the simple random walk on it, that are preserved by random perturbations. To address problems raised by those authors, we study simple random walk on the infinite percolation cluster in Cayley graphs of certain amenable groups known as "lamplighter groups''.We prove that zero speed for random walk on a lamplighter group implies zero speed for random walk on an infinite cluster, for any supercritical percolation parameter $p$. For $p$ large enough, we also establish the converse. We prove that if $G$ has a positive anchored expansion constant then so does every infinite cluster of independent percolation with parameter $p$ sufficiently close to 1; We also show that positivity of the anchored expansion constant is preserved under a random stretch if, and only if, the stretching law has an exponential tail.

Large clusters in supercritical percolation.

The statistical behavior of the size of large finite clusters in supercritical percolation on a finite lattice is investigated (below the critical dimension of the space d(c)=6). For this purpose, an approximate system of ordinary differential equations for a number of finite clusters is obtained. The correlation between the critical exponents zeta that determine the cluster decay law (ln n(s) approximately -s(zeta)) and the surface of clusters is shown. It is found that for clusters without self-intersections having a maximal surface zeta=1. For clusters with a small number of self-intersections zeta=1-eta. Here eta is a function depending on the ratio of the surface area of a cluster to its size, which tends to zero, when the surface tends to a maximum. For compact clusters with a minimum or near-minimum surface area, the first correction to the cluster decay law above percolation threshold (ln n(s) approximately -s((d-1)/d)) has been found on the basis of the drop model and the derived system of equations. The predictions are tested numerically on two- and three-dimensional lattices by Monte Carlo simulations. The results of the work allow one to conclude that above the percolation threshold majority of large clusters are compact and that the cluster surface is the main factor affecting its behavior in supercritical percolation.

On Large Isolated Regions in Supercritical Percolation

We consider supercritical vertex percolation in \(\mathbb{Z}^d \)d with any non-degenerate uniform oriented pattern of connection. In particular, our results apply to the more special unoriented case. We estimate the probability that a large region is isolated from ∞. In particular, “pancakes” with a radius r→∞ and constant thickness, parallel to a constant linear subspace L, are isolated with probability, whose logarithm grows asymptotically as ≍rdim(L) if percolation is possible across L and as ≍rdim(L)−1 otherwise. Also we estimate probabilities of large deviations in invariant measures of some cellular automata.

Vertex ordering and partitioning problems for random spatial graphs

Given an ordering of the vertices of a finite graph, let the induced weight for an edge be the separation of its endpoints in the ordering. Layout problems involve choosing the ordering to minimize a cost functional such as the sum or maximum of the edge weights. We give growth rates for the costs of some of these problems on supercritical percolation processes and supercritical random geometric graphs, obtained by placing vertices randomly in the unit cube and joining them whenever at most some threshold distance apart.

Large Deviations of the Finite Cluster Shape for Two-Dimensional Percolation in the Hausdorff and L1 Metric

We consider supercritical two-dimensional Bernoulli percolation. Conditionally on the event that the open cluster C containing the origin is finite, we prove that: the laws of C/N satisfy a large deviations principle with respect to the Hausdorff metric; let f(N) be a function from $${\mathbb{N}}$$ to $${\mathbb{R}}$$ such that f(N)/ln N→+∞ and f(N)/N→0 as N goes to ∞ the laws of {x∈ $${\mathbb{R}}$$ 2 : d(x, C)≤f(N)}/N satisfy a large deviations principle with respect to the L 1 metric associated to the planer Lebesgue measure. We link the second large deviations principle with the Wulff construction.

Stability of infinite clusters in supercritical percolation

. A recent theorem by Haggstrom and Peres concerning independent percolation is extended to all the quasi-transitive graphs. This theorem states that if 0<p1<p2≤1 and percolation occurs at level p1, then every infinite cluster at level p2 contains some infinite cluster at level p1. Consequences are the continuity of the percolation probability above the percolation threshold and the monotonicity of the uniqueness of the infinite cluster, i.e., if at level p1 there is a unique infinite cluster then the same holds at level p2. These results are further generalized to graphs with a “uniform percolation” property. The threshold for uniqueness of the infinite cluster is characterized in terms of connectivities between large balls.

Unpredictable paths and percolation

4 We construct a nearest-neighbor process ${S_n}$ on Z that is less predictable than simple random walk, in the sense that given the process until time $n$, the conditional probability that $S_{n+k} = x$ is uniformly bounded by $Ck^{-\infty}$ for some $\alpha > 1/2$. From this process, we obtain a probability measure $\mu$ on oriented paths in $\mathbf{Z}^3$ such that the number of intersections of two paths, chosen independently according to $\mu$, has an exponential tail. (For $d \geq 4$, the uniform measure on oriented paths from the origin in $\mathbf{Z}^d$ has this property.) We show that on any graph where such a measure on paths exists, oriented percolation clusters are transient if the retention parameter $p$ is close enough to 1. This yields an extension of a theorem of Grimmett, Kesten and Zhang, who proved that supercritical percolation clusters in $\mathbf{Z}^d$ are transient for all $d \geq 3$.

Dynamical Percolation

We study bond percolation evolving in time in such a way that the edges turn on and off independently according to a continuous time stationary 2-state Markov chain. Asking whether an infinite open cluster exists for a.e. t reduces (by Fubini's Theorem) to ordinary bond percolation. We ask whether “a.e. t” can be replaced by “every t” and show that for sub- and supercritical percolation the answer is yes (for any graph), while at criticality the answer is no for certain graphs. For instance, there exist graphs which do not percolate at criticality for a.e. t, but do percolate for some exceptional t. We show that for ℤ d , d 9, there is a.s. no infinite open cluster for all t at criticality. We give a sharp criterion for a general tree to have an infinite open cluster for some t, in terms of the effective conductance of the tree (analogous to a criterion of R. Lyons for ordinary percolation on trees). Finally, we compute the Hausdorff dimension of the set of times for which an infinite open cluster exists on a spherically symmetric tree.

The supercritical phase of percolation is well behaved

We prove a general result concerning the critical probabilities of subsets of a lattice L . It is a consequence of this result that the critical probability of a percolation process on L equals the limit of the critical probability of a slice of L as the thickness of the slice tends to infinity. This verification of one of the standard hypotheses of the subject settles many questions concerning supercritical percolation.