We examine the percolation model on $\mathbb{Z}^d$ by an approach involving lattice animals and their surface-area-to-volume ratio. For $β\in [0,2(d-1))$, let $f(β)$ be the asymptotic exponential rate in the number of edges of the number of lattice animals containing the origin which have surface-area-to-volume ratio $β$. The function $f$ is bounded above by a function which may be written in an explicit form. For low values of $β$ ($β\leq 1/p_c - 1$), equality holds, as originally demonstrated by F.Delyon. For higher values ($β> 1/p_c - 1$), the inequality is strict.
We introduce two critical exponents, one of which describes how quickly $f$ falls away from the explicit form as $β$ rises from $1/p_c - 1$, and the second of which describes how large clusters appear in the marginally subcritical regime of the percolation model. We demonstrate that the pair of exponents must satisfy certain inequalities, while other such inequalities yield sufficient conditions for the absence of an infinite cluster at the critical value. The first exponent is related to one of a more conventional nature in the scaling theory of percolation, that of correlation size. In deriving this relation, we find that there are two possible behaviours, depending on the value of the first exponent, for the typical surface-area-to-volume ratio of an unusually large cluster in the marginally subcritical regime.
We examine the percolation model in $\mathbb{Z}^d$ by an approach involving lattice animals, in which their relevant characteristic is surface-area-to-volume ratio. Two critical exponents are introduced. The first is related to the growth rate in size of the number of lattice animals up to translation whose surface-area-to-volume ratio is marginally greater than $1/p_c -1$. The second describes how unusually large clusters form in the percolation model at parameter values slightly below $p_c$. Certain inequalities on the pair of exponents cannot be satisfied, while others imply the continuity …
We introduce a formula for translating any upper bound on the percolation threshold of a lattice \g into a lower bound on the exponential growth rate of lattice animals $a(G)$ and vice-versa. We exploit this to improve on the best known asymptotic bounds on $a(\mathbb{Z}^d)$ as $d\to \infty$. Our formula remains valid if instead of lattice animals we enumerate certain sub-species called interfaces. Enumerating interfaces leads to functional duality formulas that are tightly connected to percolation and are not valid for lattice animals, as well as to strict inequalities for the percolation thr…
We introduce a method for translating any upper bound on the percolation threshold of a lattice $G$ into a lower bound on the exponential growth rate $a(G)$ of lattice animals and vice-versa. We exploit this in both directions. We improve on the best known asymptotic lower and upper bounds on $a(\mathbb{Z}^d)$ as $d\to \infty$. We use percolation as a tool to obtain the latter, and conversely we use the former to obtain lower bounds on $p_c(\mathbb{Z}^d)$. We obtain the rigorous lower bound $\dot{p}_c(\mathbb{Z}^3)>0.2522$ for 3-dimensional site percolation.
The lace expansion is a powerful perturbative technique to analyze the critical behavior of random spatial processes such as the self-avoiding walk, percolation and lattice trees and animals. The non-backtracking lace expansion (NoBLE) is a modification that allows us to improve its applicability in the nearest-neighbor setting on the $\mathbb Z^d$-lattice for percolation, lattice trees and lattice animals. The NoBLE gives rise to a recursive formula that we study in this paper at a general level. We state assumptions that guarantee that the solution of this recursive formula satisfies the inf…
A directed animal is a percolation cluster in the directed site percolation model. The aim of this paper is to exhibit a strong relation between the problem of computing the generating function $\G$ of directed animals on the square lattice, counted according to the area and the perimeter, and the problem of solving a system of quadratic equations involving unknown matrices. We present some solid evidence that some infinite explicit matrices, the fixed points of a rewriting like system are the natural solutions to this system of equations: some strong evidence is given that the problem of find…
The problem of percolation along sites of square lattice is studied. The number of contours being external boundaries for finite clusters has been estimated using geometric considerations. This estimation makes it possible to determine more precisely the upper limit for the percolation threshold.
The more exact upper estimate of the percolation threshold for the {\it site problem} on the quadratic lattice ${\Bbb Z}^2$ have been found on the basis of the cluster decomposition. It is done by the number estimate of cycles on ${\Bbb Z}^2$ which maybe external boundaries of finite clusters
The self-destructive percolation model is defined as follows: Consider percolation with parameter $p > p_c$. Remove the infinite occupied cluster. Finally, give each vertex (or, for bond percolation, each edge) that at this stage is vacant, an extra chance $δ$ to become occupied. Let $δ_c(p)$ be the minimal value of $δ$, needed to obtain an infinite occupied cluster in the final configuration. This model was introduced some years ago by van den Berg and Brouwer. They showed that, for the site model on the square lattice (and a few other 2D lattices satisfying a special technical condition) tha…
The Euclidean first-passage percolation model of Howard and Newman is a rotationally invariant percolation model built on a Poisson point process. It is known that the passage time between 0 and $ne_1$ obeys a diffusive upper bound: $\mbox{Var}\, T(0,ne_1) \leq Cn$, and in this paper we improve this inequality to $Cn/\log n$. The methods follow the strategy used for sublinear variance proofs on the lattice, using the Falik-Samorodnitsky inequality and a Bernoulli encoding, but with substantial technical difficulties. To deal with the different setup of the Euclidean model, we represent the pas…
We prove that the growth constants for nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice Zd are asymptotic to 2de as the dimension goes to infinity, and that their critical one-point functions converge to e. Similar results are obtained in dimensions d>8 in the limit of increasingly spread-out models; in this case the result for the growth constant is a special case of previous results of M. Penrose. The proof is elementary, once we apply previous results of T. Hara and G. Slade obtained using the lace expansion.
We compute the first three terms of the 1/d expansions for the growth constants and one-point functions of nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice Zd, with rigorous error estimates. The proof uses the lace expansion, together with a new expansion for the one-point functions based on inclusion-exclusion.
Assign to each site of the integer lattice $\Zd$ a real score, sampled according to the same distribution $F$, independently of the choices made at all other sites. A lattice animal is a finite connected set of sites, with its weight being the sum of the scores at its sites. Let $N_n$ be the maximal weight of those lattice animals of size $n$ that contain the origin. Denote by $N$ the almost sure finite constant limit of $n^{-1} N_n$, which exists under a mild condition on the positive tail of $F$. We study certain geometrical aspects of the lattice animal with maximal weight among those conta…
This paper surveys the results of recent collaborations with Eric Derbez and with Takashi Hara, which show that intergrated super-Brownian excursion (ISE) arises as the scaling limit of both lattice trees and the incipient infinite percolation cluster, in high dimensions. A potential extension to oriented percolation is also mentioned.
In Poisson percolation each edge becomes open after an independent exponentially distributed time with rate that decreases in the distance from the origin. As a sequel to our work on the square lattice, we describe the limiting shape of the component containing the origin in the oriented case. We show that the density of occupied sites at height $y$ in the cluster is close to the percolation probability in the corresponding homogeneous percolation process, and we study the fluctuations of the boundary.
In this paper, we consider a generalization of percolation: percolation of three related fluids on a honeycomb lattice. K. Izyurov and A. Magazinov proved that percolations of distinct fluids between opposite sides on a fixed hexagon become mutually independent as the lattice step tends to 0. This paper exposes this proof in details (with minor simplifications) for nonspecialists. In addition, we state a few related conjectures based on numerical experiments.
In this paper we consider independent site percolation in a triangulation of $\mathbb{R}^2$ given by adding $\sqrt{2}$-long diagonals to the usual graph $\mathbb{Z}^2$. We conjecture that $p_c=\frac{1}{2}$ for any such graph, and prove it for almost every such graph.
We consider a dilute lattice obtained from the usual $\mathbb{Z}^3$ lattice by removing independently each of its columns with probability $1-ρ$. In the remaining dilute lattice independent Bernoulli bond percolation with parameter $p$ is performed. Let $ρ\mapsto p_c(ρ)$ be the critical curve which divides the subcritical and supercritical phases. We study the behavior of this curve near the disconnection threshold $ρ_c = p_c^{\text{site}}(\mathbb{Z}^2)$ and prove that, uniformly over $ρ$ it remains strictly below $1/2$ (the critical point for bond percolation on the square lattice $\mathbb{Z}…
The understanding of site percolation on the triangular lattice progressed greatly in the last decade. Smirnov proved conformal invariance of critical percolation, thus paving the way for the construction of its scaling limit. Recently, the scaling limit of near-critical percolation was also constructed by Garban, Pete and Schramm. The aim of this very modest contribution is to explain how these results imply the convergence, as p tends to p_c, of the Wulff crystal to a Euclidean disk. The main ingredient of the proof is the rotational invariance of the scaling limit of near-critical percolati…
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.
This paper studies the first passage percolation (FPP) model: each edge in the cubic lattice is assigned a random passage time, and consideration is given to the behavior of the percolation region $B(t)$, which consists of those vertices that can be reached from the origin within a time $t > 0$. Cox and Durrett showed the shape theorem for the percolation region, saying that the normalized region $B(t)/t$ converges to some limit shape $\mathcal{B}$. This paper introduces a general FPP model defined on crystal lattices, and shows the monotonicity of the limit shapes under covering maps, thereby…
We prove that the critical probability for the Sierpinski carpet lattice in two dimensions is uniquely determined. The transition is sharp. This extends the Kumagai's result to the original Sierpinski carpet lattice.
In this paper we consider a connected planar graph $G$ and impose conditions that results in $G$ having a percolation lattice-like cellular structure. Assigning each cell of $G$ to be either occupied or vacant, we describe the outermost boundaries of star and plus connected components in $G$. We then consider the dual graph of $G$ and impose conditions under which the dual is also a percolation lattice. Finally, using $G$ and its dual, we construct vacant cell cycles surrounding occupied components and study left right crossings and bond percolation in rectangles.
We study long-range percolation on the hierarchical lattice of order $N$, where any edge of length $k$ is present with probability $p_k=1-\exp(-β^{-k} α)$, independently of all other edges. For fixed $β$, we show that the critical value $α_c(β)$ is non-trivial if and only if $N < β< N^2$. Furthermore, we show uniqueness of the infinite component and continuity of the percolation probability and of $α_c(β)$ as a function of $β$. This means that the phase diagram of this model is well understood.
Zhang found a simple, elegant argument deducing the non-existence of an infinite open cluster in certain lattice percolation models (for example, p=1/2 bond percolation on the square lattice) from general results on the uniqueness of an infinite open cluster when it exists; this argument requires some symmetry. Here we show that a simple modification of Zhang's argument requires only 2-fold (or 3-fold) symmetry, proving that the critical probabilities for percolation on dual planar lattices with such symmetry sum to 1. Like Zhang's argument, our extension applies in many contexts; in particula…
Kesten showed the exponential decay of percolation probability in the subcritical phase for the two-dimensional percolation model. This result implies his celebrated computation that $p_c=0.5$ for bond percolation in the square lattice, and site percolation in the triangular lattice, respectively. In this paper, we present a simpler proof for Kesten's theorem.
In the literature, most of the results about the enumeration of directed animals on lattices via gas considerations are obtained by a formal passage to the limit of enumeration of directed animals on cyclical versions of the lattice. Here we provide a new point of view on this phenomenon. Using the gas construction given in [Electron. J. Combin. (2007) 14 R71], we describe the gas process on the cyclical versions of the lattices as a cyclical Markov chain (roughly speaking, Markov chains conditioned to come back to their starting point). Then we introduce a notion of convergence of graphs, suc…
In this paper we present the proof of the convergence of the critical bond percolation exploration process on the square lattice to the trace of SLE$_{6}$. This is an important conjecture in mathematical physics and probability. The case of critical site percolation on the hexagonal lattice was established in the seminal work of Smirnov via proving Cardy's formula. Our proof uses a series of transformations and conditioning to construct a pair of paths: the $+\partial$CBP and the $-\partial$CBP. The convergence in the site percolation case on the hexagonal lattice allows us to obtain certain e…
For a model of a driven interface in an elastic medium with random obstacles we prove existence of a stationary positive supersolution at non-vanishing driving force. This shows the emergence of a rate independent hysteresis through the interaction of the interface with the obstacles, despite a linear (force=velocity) microscopic kinetic relation. We also prove a percolation result, namely the possibility to embed the graph of an only logarithmically growing function in a next-nearest neighbor site-percolation cluster at a non-trivial percolation threshold.
Bootstrap percolation on a graph is a deterministic process that iteratively enlarges a set of occupied sites by adjoining points with at least $θ$ occupied neighbors. The initially occupied set is random, given by a uniform product measure with a low density $p$. Our main focus is on this process on the product graph $\mathbb{Z}^2\times K_n^2$, where $K_n$ is a complete graph. We investigate how $p$ scales with $n$ so that a typical site is eventually occupied. Under critical scaling, the dynamics with even $θ$ exhibits a sharp phase transition, while odd $θ$ yields a gradual percolation tran…
The aim of this paper is to explore possible ways of extending Smirnov's proof of Cardy's formula for critical site-percolation on the triangular lattice to other cases (such as bond-percolation on the square lattice); the main question we address is that of the choice of the lattice embedding into the plane which gives rise to conformal invariance in the scaling limit. Even though we were not able to produce a complete proof, we believe that the ideas presented here go in the right direction.
We study the effect of positive correlations on the critical threshold of site and bond percolation in a square lattice with d = 2. We propose two algorithms for generating dependent lattices with minimal correlation length and non-negative n-point correlations whose critical behavior is then compared with that of independent lattices. For site percolation, we show numerically that the introduction of this specific form of positive correlation results in a lower percolation threshold, i.e., higher connectivity. For bond percolation, the opposite is observed. In this case, however, we show that…
We propose a novel probabilistic method for detection of objects in noisy images. The method uses results from percolation and random graph theories. We present an algorithm that allows to detect objects of unknown shapes in the presence of random noise. The algorithm has linear complexity and exponential accuracy and is appropriate for real-time systems. We prove results on consistency and algorithmic complexity of our procedure.
