Critical Probability For Site Percolation Research Articles

On Equivalence between Critical Probabilities of Dynamic Gossip Protocol and Static Site Percolation

The relationship between the critical probability of gossip protocol on the square lattice and the critical probability of site percolation on the square lattice is discussed. Specifically, these two critical probabilities are analytically shown to be equal to each other. Furthermore, we present a way of evaluating the critical probability of site percolation by approximating the saturation of gossip protocol. Finally, we provide numerical results which support the theoretical analysis.

Line-of-Sight Percolation

Given ω ≥ 1, let $\Z^2_{(\omega)}$ be the graph with vertex set $\Z^2$ in which two vertices are joined if they agree in one coordinate and differ by at most ω in the other. (Thus $\Z^2_{(1)}$ is precisely $\Z^2$.) Let pc(ω) be the critical probability for site percolation on $\Z^2_{(\omega)}$. Extending recent results of Frieze, Kleinberg, Ravi and Debany, we show that limω→∞ωpc(ω)=log(3/2). We also prove analogues of this result for the n-by-n grid and in higher dimensions, the latter involving interesting connections to Gilbert's continuum percolation model. To prove our results, we explore the component of the origin in a certain non-standard way, and show that this exploration is well approximated by a certain branching random walk.

Self-Organized Forest-Fires Near the Critical Time

We consider a forest-fire model which, somewhat informally, is described as follows: Each site (vertex) of the square lattice is either vacant or occupied by a tree. Vacant sites become occupied at rate 1. Further, each site is hit by lightning at rate λ. This lightning instantaneously destroys (makes vacant) the occupied cluster of the site. This model is closely related to the Drossel-Schwabl forest-fire model, which has received much attention in the physics literature. The most interesting behaviour seems to occur when the lightning rate goes to zero. In the physics literature it is believed that then the system has so-called self-organized critical behaviour. We let the system start with all sites vacant and study, for positive but small λ, the behaviour near the ‘critical time’ t c , defined by the relation 1 − exp(− t c ) = p c , the critical probability for site percolation. Intuitively one might expect that if, for fixed t > t c , we let simultaneously λ tend to 0 and m to ∞, the probability that some tree at distance smaller than m from O is burnt before time t goes to 1. However, we show that under a percolation-like assumption (which we can not prove but believe to be true) this intuition is false. We compare with the case where the square lattice is replaced by the directed binary tree, and pose some natural open problems.

On AB Bond Percolation on the Square Lattice and AB Site Percolation on Its Line Graph

We prove that AB site percolation occurs on the line graph of the square lattice when \(p \in (1 - \sqrt {1 - p_c } ,\sqrt {1 - p_c } )\), where p c is the critical probability for site percolation in \(\mathbb{Z}^2\). Also, we prove that AB bond percolation does not occur on \(\mathbb{Z}^2\) for p = \(\frac{1}{2}\).

A central limit theorem for “critical” first-passage percolation in two dimensions

Consider (independent) first-passage percolation on the edges of ℤ 2 . Denote the passage time of the edge e in ℤ 2 by t(e), and assume that P{t(e) = 0} = 1/2, P{0<t(e)<C 0 } = 0 for some constant C 0 >0 and that E[t δ (e)]<∞ for some δ>4. Denote by b 0,n the passage time from 0 to the halfplane {(x,y): x ≧ n}, and by T( 0 ,nu) the passage time from 0 to the nearest lattice point to nu, for u a unit vector. We prove that there exist constants 0<C 1 , C 2 <∞ and γ n such that C 1 ( log n) 1/2 ≦γ n ≦ C 2 ( log n) 1/2 and such that γ n −1 [b 0,n −Eb 0,n ] and (√ 2γ n ) −1 [T( 0 ,nu) − ET( 0 ,nu)] converge in distribution to a standard normal variable (as n →∞, u fixed). A similar result holds for the site version of first-passage percolation on ℤ 2 , when the common distribution of the passage times {t(v)} of the vertices satisfies P{t(v) = 0} = 1−P{t(v) ≧ C 0 } = p c (ℤ 2 , site ) := critical probability of site percolation on ℤ 2 , and E[t δ (u)]<∞ for some δ>4.

On the phase transition to sheet percolation in random Cantor sets

Thed-dimensional random Cantor set is a generalization of the classical “middle-thirds” Cantor set. Starting with the unit cube [0, 1] d , at every stage of the construction we divide each cube remaining intoMd equal subcubes, and select each of these at random with probabilityp. The resulting limit set is a random fractal, which may be crossed by paths or (d−1)-dimensional “sheets”. We examine the critical probabilityps(M, d) marking the existence of these sheet crossings, and show that ps(M,d)→1−pc(Md) asM→∞, where pc(Md) is the critical probability of site percolation on the lattice (Md) obtained by adding the diagonal edges to the hypercubic lattice ℤd. This result is then used to show that, at least for sufficiently large values ofM, the phases corresponding to the existence of path and sheet crossings are distinct.

A new lower bound for the critical probability of site percolation on the square lattice

The critical probability for site percolation on the square lattice is not known exactly. Several authors have given rigorous upper and lower bounds. Some recent lower bounds are (each displayed here with the first three digits) 0.503 (Tóth [13]), 0.522 (Zuev [15]), and the best lower bound so far, 0.541 (Menshikov and Pelikh [12]). By a modification of the method of Menshikov and Pelikh we get a significant improvement, namely, 0.556. Apart from a few classical results on percolation and coupling, which are explicitly stated in the Introduction, this paper is self-contained. © 1996 John Wiley & Sons, Inc.

Percolation and the hard-core lattice gas model

Recently a uniqueness condition for Gibbs measures in terms of disagreement percolation (a type of dependent percolation involving two realizations) has been obtained. In general this condition is sufficient but not necessary for uniqueness. In the present paper we study the hard-core lattice gas model which we abbreviate as hard-core model. This model is not only relevant in Statistical Physics, but was recently rediscovered in Operations Research in the context of certain communication networks. First we show that the uniqueness result mentioned above implies that the critical activity for the hard-core model on a graph is at least P c (1 − P c) , where P c is the critical probability for site percolation on that graph. Then, for the hard-core model on bi-partite graphs, we study the probability that a given vertex v is occupied under the two extreme boundary conditions, and show that the difference can be written in terms of the probability of having a ‘path of disagreement’ from v to the boundary. This is the key to a proof that, for this case, the uniqueness condition mentioned above is also necessary, i.e. roughly speaking, phase transition is equivalent with disagreement percolation in the product space. Finally, we discuss the hard-core model on Z d with two different values of the activity, one for the even, and one for the odd vertices. It appears that the question whether this model has a unique Gibbs measure, can, in analogy with the standard ferromagnetic Ising model, be reduced to the question whether the third central moment of the surplus of odd occupied vertices for a certain class of finite boxes is negative.

An Inequality for Greedy Lattice Animals

Let $\{X_\nu: \nu \in \mathbb{Z}^d\}$ be i.i.d. positive random variables with $E\{X^d_0(\log^+ X_0)^{d+\varepsilon}\} 0$ and $d \geq 2$. Define $M_n$ and $N_n$ by \begin{align*} M_n &= \max\big\{\sum_{\nu \in \pi} X_\nu: \pi \text{a self-avoiding path of length}\quad n \text{starting at the origin}\big\}, N_n &= \max\big\{\sum_{\nu\in \xi} X_\nu:\xi \quad\text{a lattice animal of size} \quad n \text{containing the origin}\big\}.\end{align*} Then it has been shown that there exist $M < \infty$ and $N < \infty$ such that $\frac{M_n}{n} \rightarrow M \quad\text{and}\quad \frac{N_n}{n} \rightarrow N \text{a.s. and in}\quad L^1$. In this paper we show that $M = N$ if and only if $X_0$ has bounded support and $P\{x_0 = R\} \geq p_c$, where $R$ is the right end point of support of $X_0$ and $p_c$ is the critical probability for site percolation on $\mathbb{Z}^d$.

A uniqueness condition for Gibbs measures, with application to the 2-dimensional Ising antiferromagnet

A uniqueness condition for Gibbs measures is given. This condition is stated in terms of (absence of) a certain type of percolation involving two independent realisations. This result can be applied in certain concrete situations by comparison with “ordinary” percolation. In this way we prove that the Ising antiferromagnet on a square lattice has a unique Gibbs measure if β(4−|h⥻)<1/2ln(P c /(1−P c )), whereh denotes the external magnetic field, β the inverse temperature, andP c the critical probability for site percolation on that lattice. SinceP c is larger than 1/2, this extends a result by Dobrushin, Kolafa and Shlosman (whose proof was computer-assisted).

On the geometry of random Cantor sets and fractal percolation

Random Cantor sets are constructions which generalize the classical Cantor set, “middle third deletion” being replaced by a random substitution in an arbitrary number of dimensions. Two results are presented here. (a) We establish a necessary and sufficient condition for the projection of ad-dimensional random Cantor set in [0,1]d onto ane-dimensional coordinate subspace to contain ane-dimensional ball with positive probability. The same condition applies to the event that the projection is the entiree-dimensional unit cube [0,1] e . This answers a question of Dekking and Meester,(9) (b) The special case of “fractal percolation” arises when the substitution is as follows: The cube [0,1] d is divided intoM d subcubes of side-lengthM −, and each such cube is retained with probabilityp independently of all other subcubes. We show that the critical valuep c(M, d) ofp, marking the existence of crossings of [0,1] d contained in the limit set, satisfiesp c(M, d)→p c(d) asM→∞, wherep c(d) is the critical probability of site percolation on a latticeL d obtained by adding certain edges to the hypercubic lattice ℤ d . This result generalizes in an unexpected way a finding of Chayes and Chayes,(4) who studied the special case whend=2.

An Upper Bound on the Critical Percolation Probability for the Three- Dimensional Cubic Lattice

We prove that the critical probability for site percolation on the three-dimensional cubic lattice satisfies the inequality $p^{(3)}_c < 1/2$. An application to the three-dimensional Ising model is given.

Combined site and bond percolation with nearest- and next-nearest-neighbour bonds

Percolation on a square lattice, in which there exist different occupation probabilities for sites, nearest-neighbour bonds and next-nearest-neighbour bonds, is studied by a real-space renormalization technique. The critical surface is controlled by a single fixed point, in accordance with universality. The critical probability for site percolation on the square lattice with diagonal bonds present, is estimated to be p ̃ c =0.41 , in agreement with series estimates.