Uniqueness and nonuniqueness in percolation theory pdf

Percolation on transitive graphs 3 recent progress in percolation theory. We only consider the problem for autonomous odes, but note that through 1. Asamodelforadisorderedmedium,percolationisoneofthesimplest,incorporating as it does a minimum. In case of nonuniqueness we say that there is a phase transition. Socalled simultaneous uniqueness in couplings of percolation processes is also considered. Nevertheless, as we shall see in section 8, certain deep results for percolation in zd uniqueness in orthants, and estimates of pc have signi. Uniqueness in percolation theory, statistica neerlandica. Under what conditions, there exists a unique solution to 1. Chalmers, mathematical sciences, mathematical statistics. Request pdf uniqueness and nonuniqueness in percolation theory this paper is an uptodate introduction to the problem of uniqueness versus nonuniqueness of infinite clusters for percolation.

Despite its very simple rules, percolation theory has successfully been applied to describe a large. Kopclman department of chemistry, the university ofmichigan, ann arbor, michigan 48109 received 1 june 1976 a new approach for the determination of the critical percolation concentration, percolation. Explicitly, for the ising magnet, the region of nonuniqueness is characterized by percolation of cspins in the cstate. Key properties of the graphs in this context turn out to be amenability and nonamenability. Combined effects of prevention and quarantine on a. In the case of electrostatics, this means that there is a unique electric field derived from a potential function satisfying poissons equation under the boundary conditions. Open cluster is a connected component of the open vertex graph.

Tree graph inequalities and critical behavior in percolation models. Uniqueness and nonuniqueness in percolation theory. It will depend also from the last thing, the kind of percolation we are studying. Rigidity percolation and boundary conditions holroyd, alexander e. Percolation on nonamenable products at the uniqueness. Percolation on transitive graphs as a coalescent process. For this purpose, we investigate the probability distribution function pdf of the size of a locallyde. Some of the main results are proved in detail, while for others the proofs are merely sketched, and for yet others they are omitted.

In this thesis, we provide a framework that extends the cluster description of cont and bouchaud by means of percolation theory 10. The applications of percolation theory to materials science and other domains are discussed in the article percolation. Equally importantly, however, the theory seeks to provide some insights into important social phenomena. Once percolation theory is defined, we explore applications to the renormalization group, computer simulations of potts models, and randomly punctured conducting sheets. Harris from the 1960s, proved the obvious result that the critical probability for bond percolation on z2 is 12. Here we give a counterexample to this in higher dimensions. The uniqueness theorem for poissons equation states that, for a large class of boundary conditions, the equation may have many solutions, but the gradient of every solution is the same. Diffusion and conduction in percolation systems theory and applications armin bunde and jan w. It is known that in one dimension an irreducible subshift of finite type has a unique measure of maximal entropy, the socalled parry measure. Uniqueness and nonuniqueness in percolation theory 2005. On non uniqueness of percolation on nonamenable cayley graphs. These clusters can correspond to mutual funds or to herding among security analysts in the context of a stock market. Fisher and kadanoff helped to develop percolation theory and understand the percolation as a critical phenomena 9fractal concept mandelbrot, 1977 new tools fractal geometry together with computer development pushed forward the percolation theory 9still many open questions exist.

In this lecture, we will discuss some problems on random walks and percolations from a viewpoint of quasiisometries, including rigidity of speed exponents and of cuttimescutpoints for random walks on finitely generated groups, and uniqueness and nonuniqueness for. Uniqueness vs nonuniqueness in complete connections with modified majority rules. Phase transition and uniqueness of levelset percolation erik broman and ronald meester august 11, 2018 abstract the main purpose of this paper is to introduce and establish basic results of a natural extension of the classical boolean percolation model also known as the gilbert disc model. In statistical physics and mathematics, percolation theory describes the behaviour of connected clusters in a random graph. We study the probability of getting more than one spanning cluster at pc for site percolation in dimensions of two to five. Site and bond percolation a site can be a field or a node of a lattice bond percolation. Cluster multiple labeling technique and critical concentration algorithm j. Phase transition and uniqueness of levelset percolation. Random walk and percolation under a quasiisometry viewpoint abstract. At this point it is appropriate to introduce the theory of uniqueness. Pdf nonuniqueness of spanning clusters in two to five. Uniqueness and nonuniqueness in percolation theory arxiv.

Some of the main results are proved in detail, while for others the proofs. Percolation theory and fire propagation in a forest author. Percolation theory isaac brodsky this essay describes percolation theory. There are two kinds of percolation, site and bond percolation. The existence and uniqueness theorem of the solution a first order linear equation initial value problem does an initial value problem always a solution. Uniqueness of the infinite cluster and continuity of connectivity functions in shortand longrange percolation. The most wellknown condition which implies uniqueness is dobrushins condition of weak dependence dobrushin 1968a. Site percolation site percolation model the space of the model is zn or any in. Russos formula, uniqueness of the infinite cluster, and continuous differentiability of free energy for continuum percolation jiang, jianping, zhang, sanguo, and guo, tiande, journal of applied probability, 2011. Chapter 4 existence and uniqueness of solutions for nonlinear odes in this chapter we consider the existence and uniqueness of solutions for the initial value problem for general nonlinear odes. The following theorem states a precise condition under which exactly one solution would always exist for. Sorry, we are unable to provide the full text but you may find it at the following locations. The same problem is considered for certain dependent percolation models most prominently.

Uniqueness and nonuniqueness in percolation theory project euclid. Under what conditions, there exists a solution to 1. Diffusion and conduction in percolation systems theory. Nonuniqueness of measures of maximal entropy for subshifts of finite type. For iid percolation on z d, uniqueness of the infinite cluster is a classical result, while on certain other transitive graphs uniqueness may fail. Communications in mathematical physics, 111 1987, 505532. Introduction to percolation theory danica stojiljkovic 11 6112010 introduction to percolations system in concern discrete system in d dimensions lattices. This construction is based on a device, already used in 3, that consists of producing sub and supersolutions for. A central problem in the theory is to determine whether a given specification has a unique gibbs measure. The existence and uniqueness theorem of the solution a. Kantelhardt 1 introduction percolation is a standard model for disordered systems. Local clustersize statistics in the critical phase of.

Bond percolation onzd is indeed a special case, but probably it exhibits the majority of properties expected of more general. Recently there has been an increased interest in twodimensional percolation mainly due to the fact that critical percolation on the. The final density of the recovered sites r for the combined model of both prevention site percolation and quarantine bond percolation with a constant infection rate. Chapter 4 existence and uniqueness of solutions for. For iid percolation on zd, uniqueness of the infinite cluster is a classical result, while on certain other transitive graphs uniqueness may fail. Percolation theory and fire propagation in a forest. In a onedimensional lattice, the percolation threshold is obviously equal to 1.

For the potts models on z2 there are characterization theorems relating the uniqueness of the gibbs states and the absence of spinsystem percolation cnpr1, cnpr2, c1. The vertices are open or closed with probability p, which may depend on the properties of the vertex e. Uniqueness in percolation theory uniqueness in percolation theory meester, r. Percolation theory an overview sciencedirect topics. Recall that it is this property that underlies the existence of a ow. Uniqueness and non uniqueness in percolation theory journal article, 2006. Percolation and disordered systems 143 preface this course aims to be a nearly selfcontained account of part of the mathematical theory of percolation and related topics. The central result of percolation theory is the existence of the critical probability pc, the percolation threshold at which a cluster of infinite size appears. We show that the pdf of the vertices in the outer layers in. Furthermore, the theory also attempts to explain and integrate a wide variety of research findings from different response domains.