The most wellknown condition which implies uniqueness is dobrushins condition of weak dependence dobrushin 1968a. Uniqueness theorem for poissons equation wikipedia. In statistical physics and mathematics, percolation theory describes the behaviour of connected clusters in a random graph. At this point it is appropriate to introduce the theory of uniqueness. For this purpose, we investigate the probability distribution function pdf of the size of a locallyde. Percolation on transitive graphs 3 recent progress in percolation theory. Bond percolation onzd is indeed a special case, but probably it exhibits the majority of properties expected of more general. We only consider the problem for autonomous odes, but note that through 1. In case of nonuniqueness we say that there is a phase transition. Percolation theory an overview sciencedirect topics.
Recently there has been an increased interest in twodimensional percolation mainly due to the fact that critical percolation on the. 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. Once percolation theory is defined, we explore applications to the renormalization group, computer simulations of potts models, and randomly punctured conducting sheets. Uniqueness and nonuniqueness in percolation theory 2005. Under what conditions, there exists a solution to 1. Furthermore, the theory also attempts to explain and integrate a wide variety of research findings from different response domains.
Site percolation site percolation model the space of the model is zn or any in. Uniqueness and nonuniqueness in percolation theory project euclid. Rigidity percolation and boundary conditions holroyd, alexander e. 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. Diffusion and conduction in percolation systems theory and applications armin bunde and jan w. Fromamathematicalperspective,thenotionofconformalinvari. 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. Recall that it is this property that underlies the existence of a ow. Uniqueness of the infinite cluster and continuity of connectivity functions in shortand longrange percolation. There are two kinds of percolation, site and bond percolation. Site and bond percolation a site can be a field or a node of a lattice bond percolation.
Open cluster is a connected component of the open vertex graph. Some of the main results are proved in detail, while for others the proofs. Local clustersize statistics in the critical phase of. Equally importantly, however, the theory seeks to provide some insights into important social phenomena. Percolation theory and fire propagation in a forest author. The existence and uniqueness theorem of the solution a. Tree graph inequalities and critical behavior in percolation models. The existence and uniqueness theorem of the solution a first order linear equation initial value problem does an initial value problem always a solution. 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. It will depend also from the last thing, the kind of percolation we are studying. Here we give a counterexample to this in higher dimensions.
Uniqueness in percolation theory, statistica neerlandica. In this thesis, we provide a framework that extends the cluster description of cont and bouchaud by means of percolation theory 10. Percolation on transitive graphs as a coalescent process. Asamodelforadisorderedmedium,percolationisoneofthesimplest,incorporating as it does a minimum. The applications of percolation theory to materials science and other domains are discussed in the article percolation. Chapter 4 existence and uniqueness of solutions for.
Some of the main results are proved in detail, while for others the proofs are merely sketched, and for yet others they are omitted. Despite its very simple rules, percolation theory has successfully been applied to describe a large. The following theorem states a precise condition under which exactly one solution would always exist for. Uniqueness and non uniqueness in percolation theory journal article, 2006.
Uniqueness vs nonuniqueness in complete connections with modified majority rules. Uniqueness and nonuniqueness in percolation theory core. These clusters can correspond to mutual funds or to herding among security analysts in the context of a stock market. Percolation theory and fire propagation in a forest. The vertices are open or closed with probability p, which may depend on the properties of the vertex e. Diffusion and conduction in percolation systems theory. Uniqueness in percolation theory uniqueness in percolation theory meester, r. Socalled simultaneous uniqueness in couplings of percolation processes is also considered. Percolation is the simplest fundamental model in statistical mechanics that exhibits phase transitions signaled by the emergence of a giant connected component. Communications in mathematical physics, 111 1987, 505532. Box 800 10 3508 ta utrecht the netherlands the possible uniqueness of the infinite cluster in percolation theory has been a subject of research for more then three decades. Sorry, we are unable to provide the full text but you may find it at the following locations. Explicitly, for the ising magnet, the region of nonuniqueness is characterized by percolation of cspins in the cstate.
For iid percolation on z d, uniqueness of the infinite cluster is a classical result, while on certain other transitive graphs uniqueness may fail. 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. 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. Uniqueness and nonuniqueness in percolation theory arxiv. A central problem in the theory is to determine whether a given specification has a unique gibbs measure. Nonuniqueness of measures of maximal entropy for subshifts of finite type. 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. Chalmers, mathematical sciences, mathematical statistics.
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 isaac brodsky this essay describes percolation theory. Introduction to percolation theory danica stojiljkovic 11 6112010 introduction to percolations system in concern discrete system in d dimensions lattices. Kantelhardt 1 introduction percolation is a standard model for disordered systems. Combined effects of prevention and quarantine on a. Key properties of the graphs in this context turn out to be amenability and nonamenability. 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. It is known that in one dimension an irreducible subshift of finite type has a unique measure of maximal entropy, the socalled parry measure. Cluster multiple labeling technique and critical concentration algorithm j. 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 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.
Percolation on nonamenable products at the uniqueness. Nevertheless, as we shall see in section 8, certain deep results for percolation in zd uniqueness in orthants, and estimates of pc have signi. This construction is based on a device, already used in 3, that consists of producing sub and supersolutions for. In a onedimensional lattice, the percolation threshold is obviously equal to 1. Random walk and percolation under a quasiisometry viewpoint abstract. On non uniqueness of percolation on nonamenable cayley graphs.
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