The critical probability for Voronoi percolation in the hyperbolic plane tends to 1/2

We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster tends to 1/2 as the intensity of the Poisson process tends to infinity. This confirms a conjecture of Benjamini and Schramm [5].     

The critical probability for confetti percolation equals 1/2

In the confetti percolation model, or two‐coloured dead leaves model, radius one disks arrive on the plane according to a space‐time Poisson process. Each disk is coloured black with probability p and white with probability . In this paper we show that the critical probability for confetti percolation equals 1/2. That is, if p > 1/2 then a.s. there is an unbounded curve in the plane all of whose points are black; while if then a.s. all connected components of the set of black points are bounded. This answers a question of Benjamini and Schramm [1]. The proof builds on earlier work by Hirsch [7] and makes use of an adaptation of a sharp thresholds result of Bourgain. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 679–697, 2017     

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     

New methods to bound the critical probability in fractal percolation

We study the critical probability p_c(M) in two‐dimensional M‐adic fractal percolation. To find lower bounds, we compare fractal percolation with site percolation. Fundamentally new is the construction of a computable increasing sequence that converges to p_c(M). We prove that and . For the upper bounds, we introduce an iterative random process on a finite alphabet , which is easier to analyze than the original process. We show that and . © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 710–730, 2015     

Improved upper bounds for the critical probability of oriented percolation in two dimensions

We refine the method of our previous paper [2] which gave upper bounds for the critical probability in two-dimensional oriented percolation. We use our refinement to show that © 1994 John Wiley & Sons, Inc.     

The survival probability for critical spread-out oriented percolation above 4 + 1 dimensions. I. Induction

We consider critical spread-out oriented percolation above 4 + 1 dimensions. Our main result is that the extinction probability at time n (i.e., the probability for the origin to be connected to the hyperplane at time n but not to the hyperplane at time n + 1) decays like 1/Bn ² as , where B is a finite positive constant. This in turn implies that the survival probability at time n (i.e., the probability that the origin is connected to the hyperplane at time n) decays like 1/Bn as . The latter has been shown in an earlier paper to have consequences for the geometry of large critical clusters and for the incipient infinite cluster. The present paper is Part I in a series of two papers. In Part II, we derive a lace expansion for the survival probability, adapted so as to deal with point-to-plane connections. This lace expansion leads to a nonlinear recursion relation for the survival probability. In Part I, we use this recursion relation to deduce the asymptotics via induction.     

Monotonicity of the probability of percolation for Bernoulli random fields on periodic graphs

In this work, the problem of percolation of the Bernoulli random field on periodic graphs ?? of an arbitrary dimension d is studied. A theorem on nondecreasing dependence of the probability of percolation Q(c ₁ , ?? , c _n ) with respect to each of the parameters c _i , i = 1÷n, ?C concentration of the Bernoulli field is proved.     

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

Summary We prove by elementary combinatorial considerations that the critical probability of the square lattice site percolation is larger than 0.503478.Work supported by the Central Research Found of the Hungarian Academy of Sciences (Grant No. 476/82)     

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 in half-spaces: equality of critical densities and continuity of the percolation probability

Renormalization arguments are developed and applied to independent nearest-neighbor percolation on various subsets of ^d ,d2, yielding:

The survival probability of a critical branching process in a Markovian random environment

In this Note, we first prove a local limit theorem for a semi-Markov chain and then apply it to study the asymptotic behavior of the survival probability of a critical branching process in Markovian random environment.     

Sharp thresholds and percolation in the plane

Recently, it was shown by Bollobás and Riordan 4 that the critical probability for random Voronoi percolation in the plane is 1/2. As a by‐product of the method, a short proof of the Harris–Kesten Theorem was given by Bollobás and Riordan 5 . The aim of this paper is to show that the techniques used in these papers can be applied to many other planar percolation models, both to obtain short proofs of known results and to prove new ones. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

The Brownian map is the scaling limit of uniform random plane quadrangulations

We prove that uniform random quadrangulations of the sphere with n faces, endowed with the usual graph distance and renormalized by n ^?1/4, converge as n → ∞ in distribution for the Gromov–Hausdorff topology to a limiting metric space. We validate a conjecture by Le Gall, by showing that the limit is (up to a scale constant) the so-called Brownian map, which was introduced by Marckert–Mokkadem and Le Gall as the most natural candidate for the scaling limit of many models of random plane maps. The proof relies strongly on the concept of geodesic stars in the map, which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere.     

On the mixing time of a simple random walk on the super critical percolation cluster

 We study the robustness under perturbations of mixing times, by studying mixing times of random walks in percolation clusters inside boxes in Z ^d . We show that for d≥2 and p>p _c (Z ^d ), the mixing time of simple random walk on the largest cluster inside is Θ(n ² ) – thus the mixing time is robust up to a constant factor. The mixing time bound utilizes the Lovàsz-Kannan average conductance method. This is the first non-trivial application of this method which yields a tight result. Received: 16 December 2001 / Revised version: 13 August 2002 / Published online: 19 December 2002     

The probability that random algebraic integers are relatively r-prime

Benkoski (1976) [1] proved that the probability that k randomly chosen integers do not have a nontrivial common rth power is 1/ζ(rk). We first give a more concise proof of this result before proceeding to establish its analogue in the ring of algebraic integers.     

Exchangeable random measures in the plane

A random measure on [0,1]², [0, 1]}₊ or ₊ ² is said to be separately exchangeable, if its distribution is invariant under arbitrary Lebesgue measure-preserving transformations in the two coordinates, and jointly exchangeable if is defined on [0,1]² or ₊ ² , and its distribution is invariant under mappings by a common measure-preserving transformation in both directions. In each case, we derive a general representation of in terms of independent Poisson processes and i.i.d. random variables.