The expected number of critical percolation clusters intersecting a line segment

We study critical percolation on a regular planar lattice. Let $E_G(n)$ be the expected number of open clusters intersecting or hitting the line segment $[0,n]$. (For the subscript $G$ we either take $\mathbb{H}$, when we restrict to the upper halfplane, or $\mathbb{C}$, when we consider the full lattice). Cardy (2001) (see also Yu, Saleur and Haas (2008)) derived heuristically that $E_{\mathbb{H}}(n) = An + \frac{\sqrt{3}}{4\pi}\log(n) + o(\log(n))$, where $A$ is some constant. Recently Kov\'{a}cs, Igl\'{o}i and Cardy (2012) derived heuristically (as a special case of a more general formula) that a similar result holds for $E_{\mathbb{C}}(n)$ with the constant $\frac{\sqrt{3}}{4\pi}$ replaced by $\frac{5\sqrt{3}}{32\pi}$. In this paper we give, for site percolation on the triangular lattice, a rigorous proof for the formula of $E_{\mathbb{H}}(n)$ above, and a rigorous upper bound for the prefactor of the logarithm in the formula of $E_{\mathbb{C}}(n)$.Comment: Final version, appeared in Elect.Comm.Probab. 21 (2016

A signal-recovery system: asymptotic properties and construction of an infinite volume limit

We consider a linear sequence of `nodes', each of which can be in state 0 (`off') or 1 (`on'). Signals from outside are sent to the rightmost node and travel instantaneously as far as possible to the left along nodes which are `on'. These nodes are immediately switched off, and become on again after a recovery time. The recovery times are independent exponentially distributed random variables. We present properties for finite systems and use some of these properties to construct an infinite-volume extension, with signals `coming from infinity'. This construction is related to a question by D. Aldous and we expect that it sheds some light on, and stimulates further investigation of, that question.Comment: 16 page

Two-dimensional volume-frozen percolation: exceptional scales

We study a percolation model on the square lattice, where clusters "freeze" (stop growing) as soon as their volume (i.e. the number of sites they contain) gets larger than N, the parameter of the model. A model where clusters freeze when they reach diameter at least N was studied in earlier papers. Using volume as a way to measure the size of a cluster - instead of diameter - leads, for large N, to a quite different behavior (contrary to what happens on the binary tree, where the volume model and the diameter model are "asymptotically the same"). In particular, we show the existence of a sequence of "exceptional" length scales.Comment: 20 pages, 2 figure

On the size of the largest cluster in 2D critical percolation

We consider (near-)critical percolation on the square lattice. Let M_n be the size of the largest open cluster contained in the box [-n,n]^2, and let pi(n) be the probability that there is an open path from O to the boundary of the box. It is well-known that for all 0< a < b the probability that M_n is smaller than an^2 pi(n) and the probability that M_n is larger than bn^2 pi(n) are bounded away from 0 as n tends to infinity. It is a natural question, which arises for instance in the study of so-called frozen-percolation processes, if a similar result holds for the probability that M_n is between an^2 pi(n) and bn^2 pi(n). By a suitable partition of the box, and a careful construction involving the building blocks, we show that the answer to this question is affirmative. The `sublinearity' of 1/pi(n) appears to be essential for the argument.Comment: 12 pages, 3 figures, minor change

The gaps between the sizes of large clusters in 2D critical percolation

Consider critical bond percolation on a large 2n by 2n box on the square lattice. It is well-known that the size (i.e. number of vertices) of the largest open cluster is, with high probability, of order n^2 \pi(n), where \pi(n) denotes the probability that there is an open path from the center to the boundary of the box. The same result holds for the second-largest cluster, the third largest cluster etcetera. Jarai showed that the differences between the sizes of these clusters is, with high probability, at least of order \sqrt{n^2 \pi(n)}. Although this bound was enough for his applications (to incipient infinite clusters), he believed, but had no proof, that the differences are in fact of the same order as the cluster sizes themselves, i.e. n^2 \pi(n). Our main result is a proof that this is indeed the case.Comment: 10 page

Sublinearity of the travel-time variance for dependent first-passage percolation

Let $E$ be the set of edges of the $d$-dimensional cubic lattice $\mathbb{Z}^d$, with $d\geq2$, and let $t(e),e\in E$, be nonnegative values. The passage time from a vertex $v$ to a vertex $w$ is defined as $\inf_{\pi:v\rightarrow w}\sum_{e\in\pi}t(e)$, where the infimum is over all paths $\pi$ from $v$ to $w$, and the sum is over all edges $e$ of $\pi$. Benjamini, Kalai and Schramm [2] proved that if the $t(e)$'s are i.i.d. two-valued positive random variables, the variance of the passage time from the vertex 0 to a vertex $v$ is sublinear in the distance from 0 to $v$. This result was extended to a large class of independent, continuously distributed $t$-variables by Bena\"{\i}m and Rossignol [1]. We extend the result by Benjamini, Kalai and Schramm in a very different direction, namely to a large class of models where the $t(e)$'s are dependent. This class includes, among other interesting cases, a model studied by Higuchi and Zhang [9], where the passage time corresponds with the minimal number of sign changes in a subcritical "Ising landscape."Comment: Published in at http://dx.doi.org/10.1214/10-AOP631 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Near-critical percolation with heavy-tailed impurities, forest fires and frozen percolation

Consider critical site percolation on a "nice" planar lattice: each vertex is occupied with probability $p = p_c$, and vacant with probability $1 - p_c$. Now, suppose that additional vacancies ("holes", or "impurities") are created, independently, with some small probability, i.e. the parameter $p_c$ is replaced by $p_c - \varepsilon$, for some small $\varepsilon > 0$. A celebrated result by Kesten says, informally speaking, that on scales below the characteristic length $L(p_c - \varepsilon)$, the connection probabilities remain of the same order as before. We prove a substantial and subtle generalization to the case where the impurities are not only microscopic, but allowed to be "mesoscopic". This generalization, which is also interesting in itself, was motivated by our study of models of forest fires (or epidemics). In these models, all vertices are initially vacant, and then become occupied at rate $1$. If an occupied vertex is hit by lightning, which occurs at a (typically very small) rate $\zeta$, its entire occupied cluster burns immediately, so that all its vertices become vacant. Our results for percolation with impurities turn out to be crucial for analyzing the behavior of these forest fire models near and beyond the critical time (i.e. the time after which, in a forest without fires, an infinite cluster of trees emerges). In particular, we prove (so far, for the case when burnt trees do not recover) the existence of a sequence of "exceptional scales" (functions of $\zeta$). For forests on boxes with such side lengths, the impact of fires does not vanish in the limit as $\zeta \searrow 0$.Comment: 67 pages, 15 figures (some small corrections and improvements, one additional figure); version to be submitte

Some Conditional Correlation Inequalities for Percolation and Related Processes

Consider ordinary bond percolation on a finite or countably infinite graph. Let s, t, a and b be vertices. An earlier paper proved the (nonintuitive) result that, conditioned on the event that there is no open path from s to t, the two events "there is an open path from s to a" and "there is an open path from s to b" are positively correlated. In the present paper we further investigate and generalize the theorem of which this result was a consequence. This leads to results saying, informally, that, with the above conditioning, the open cluster of s is conditionally positively (self-)associated and that it is conditionally negatively correlated with the open cluster of t. We also present analogues of some of our results for (a) random-cluster measures, and (b) directed percolation and contact processes, and observe that the latter lead to improvements of some of the results in a paper of Belitsky, Ferrari, Konno and Liggett (1997)