A new look at contact processes in several dimensions. Non uniqueness of measures of maximal entropy for subshifts of finite type volume 14 issue 2. We show that there are in general 2 n or 2 n1 equivalent sets of parameters to fit an experimental curve comprising n resonant features, of vibrational or electronic origin for example. Nonuniqueness of measures of maximal entropy for subshifts. For more about uniqueness and non uniqueness of infinite clusters in percolation theory see the survey. What is an accessible introduction to percolation theory. We prove that percolation on any such graph has a nonempty phase in which there are infinite light clusters, which implies the existence of a nonempty phase in which there are infinitely many infinite clusters. Experimental data from secondorder nonlinear optical spectroscopies sfg, dfg, and shg provide parameters relevant to the physical chemistry of interfaces and thin films.
Uniqueness in percolation theory, statistica neerlandica. A central problem in the theory is to determine if a given specification has a unique gibbs measure. It also discusses concepts like 1dependent percolation which, i feel, can be much more useful for computer scientists than independent percolation and random voronoi percolation, which is a fascinating topic, not least because it is a site percolation model whose critical probability in the plane is. Percolation theory deals with the numbers and properties of the clusters formed when sites are occupied with probability p, see fig.
Citeseerx citation query percolation on transitive. It also discusses concepts like 1dependent percolation which, i feel, can be much more useful for computer scientists than independent percolation and random voronoi percolation, which is a fascinating topic, not least because it is a site percolation model whose critical probability in the plane is 12. Uniqueness and nonuniqueness in percolation theory arxiv. Let pc denote the bond percolation threshold in zd, namely the value of p. Invariant percolation and harmonic dirichlet functions. Open cluster is a connected component of the open vertex graph. In 1981, newman and schulman ns1 proved that with probability one. Uniqueness and nonuniqueness in percolation theory. On nonuniqueness of percolation on nonamenable cayley graphs. Results concerning uniqueness of infinite clusters and inequalities for the critical value pc are given, and a simple planar example exhibiting uniqueness and non uniqueness for. A uniqueness condition for gibbs measures, with application. In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are removed. Uniqueness of uniform random colorings of regular trees.
Percolation and the existence of the infinite open cluster 3 the calculation of p c for the 2dimensional lattice requires some unique properties of the lattice. The water droplets were continuously generated from the cusp shown in figure 16. The vertices are open or closed with probability p, which may depend on the properties of the vertex e. We study bernoulli bond percolation on nonunimodular quasitransitive graphs, and more generally graphs whose automorphism group has a nonunimodular quasitransitive subgroup.
Percolation theory refers to a class of models that describe the properties of a system given the networking among its constituents. Citeseerx uniqueness and nonuniqueness in percolation theory. Nonuniqueness and meanfield criticality for percolation. This gives induced load 1 at all vertices except for the marked one, which has induced load 0. In this in nite 3regular tree, start by assigning the load of each edge to the vertex that is furthest from the marked vertex. 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. Percolation beyond zd, many questions and a few answers.
Request pdf uniqueness and non uniqueness in percolation theory this paper is an uptodate introduction to the problem of uniqueness versus non uniqueness of infinite clusters for percolation. The infinity of the space of sites or vertices and its geometry are principal features of this model, see for example 1, 2. This is a geometric type of phase transition, since at a critical fraction of removal the network breaks into significantly smaller connected clusters. Percolation theory describes connectivity of objects within a network structure, and the effects of this connectivity on the macroscale properties of the system. Percolation on transitive graphs 3 recent progress in percolation theory. A central problem in the theory is to determine whether a given specification has a unique gibbs measure. Uniqueness of the infinite cluster and continuity of. Percolation theory in the euclidean lattice zd started with the work of broadbent and hammersley in 1957. Percolation theory isaac brodsky this essay describes percolation theory.
Nov 07, 2017 we study bernoulli bond percolation on nonunimodular quasitransitive graphs, and more generally graphs whose automorphism group has a nonunimodular quasitransitive subgroup. Gtd15 in orbit equivalence theory, especially concerning the approxima. It is a cornerstone of the theory of spatial stochastic processes with applications in such fields as statistical physics, epidemiology, and the spread of populations. The aim of this note is to prove a weak version of the conjecture of benjamini and schramm about phase of non uniqueness for the bernoulli bond percolation on nonamenable transitive graphs.
Percolation is pivotal for studying more complex physical systems exhibiting phase transitions. Percolation on transitive graphs as a coalescent process. The next theorem relates the nonuniqueness of the incipient infinite cluster inside. Pdf nonuniqueness of parameters extracted from resonant. Uniqueness in percolation theory, statistica neerlandica 10. Uniqueness in percolation theory uniqueness in percolation theory meester, r. For some probability there will be a subgraph with an in. New relations are derived here between these three properties and used for the resolution of some old problems in the mathematical analysis of. Nonuniqueness and meanfield criticality for percolation on. Available formats pdf please select a format to send. Nevertheless, as we shall see in section 8, certain deep results for percolation in zd uniqueness in orthants, and estimates of pc have signi.
Non uniqueness of measures of maximal entropy for subshifts of finite type volume 14 issue 2 skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. The droplets were very effective in reducing the viscosity of the silicone oil. Percolation plays a pivotal role in studying more complex systems exhibiting phase. The burtonkeaneapproachto uniqueness is sketched in section 2 in the context of percolation. Theorem 3 frozen percolation is not almost sure unique let u. Relentless merging followed by simultaneous uniqueness. After a few days this collcction of drops reached a percolation threshold with rings, like wedding bands, of water around the rod. On non uniqueness of percolation on nonamenable cayley graphs. The constraineddegree percolation model doesnt have the insertion tolerance property. Percolation and the existence of the infinite open cluster riley heckel abstract. For percolation on a ct, the same parameter region is a critical phase 3, where the cluster size histogram exhibits a powerlaw distribution.
Monotonicity of uniqueness for percolation on cayley graphs. This paper is an uptodate introduction to the problem of uniqueness versus non uniqueness of infinite clusters for percolation on. For iid percolation on z d, uniqueness of the infinite cluster is a classical result, while on certain other transitive graphs uniqueness may fail. Sorry, we are unable to provide the full text but you may find it at the following locations. The ima volumes in mathematics and its applications, vol. Citeseerx citation query percolation on transitive graphs. Additive and cancellative interacting particle systems. Results concerning uniqueness of infinite clusters and inequalities for the critical value pc are given, and a simple planar example exhibiting uniqueness and non uniqueness for different p pc is analyzed. Write autg for the group of graph automorphisms of the graph g. He wrote in the earlier paper 21 of uniqueness in longrange percolation. Percolation theory an overview sciencedirect topics. Edges in the grid are open or present with probability p 0. Gtd15 in orbit equivalence theory, especially concerning the approxima tion of p.
Percolation theory j w essam department of mathematics, westfield college, university of london, kidderpore avenue, london nw3 7st, uk abstract the theory of percolation models is developed following general ideas in the area of critical phenomena. Uniqueness and nonuniqueness in percolation theory nasaads. Studying large networks via local weak limit theory. Percolation theory is the study of an idealized random medium in two or more dimensions.
Independent and dependent percolation jt chayes,y a. Percolation thresholds were erst studied formally by broadbent and hammersley 1957 see also ham. Percolation is a model used to explain how neural activity is transmitted across the various connecti. The applications of percolation theory to materials science and in many other disciplines are discussed here and in. Uniqueness and nonuniqueness in percolation theory core. Aizenman et al and burton and keane that there is only one, if any, infinite. The mathematical theory is mature, and continues to give rise to problems of special beauty and difficulty. Russos formula, uniqueness of the infinite cluster, and. Site percolation site percolation model the space of the model is zn or any in.
Uniqueness and nonuniqueness in percolation theory project euclid. Citeseerx scientific documents that cite the following paper. Uniqueness and nonuniqueness in percolation theory request pdf. Russos formula, uniqueness of the infinite cluster, and continuous differentiability of free energy for continuum percolation volume 48 issue 3 jianping jiang, sanguo zhang, tiande guo. Beavers university of minnesota, minneapolis 55455 received 19 april 1983 and in revised form 20 october 1983 zyxw highviscosity. Load percolation as nonuniqueness in the limit an in nite sparse graph can exhibit nonuniqueness in its balanced allocations. Nonuniqueness and meanfield criticality for percolation on nonunimodular transitive graphs.
Percolation theory is a branch of probability theory describing connectedness in a stochastic network. Elementary properties of the percolation function 9 3. Percolation in 2dsquare lattice of linear size l 5. 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. This paper is an uptodate introduction to the problem of uniqueness versus non uniqueness of infinite clusters for percolation on z d and, more generally, on transitive graphs.
Frozen percolation on the binary tree is nonendogenous. A key focus of percolation models is the study of critical phenomena, which describe the sharp transition in the behavior of a system upon the formation of a spanning network structure. Percolation is a theoretical model used to understand the way activation and diffusion of neural activity occurs within neural networks. In the lattice above, we have one cluster of size 7, a cluster of size 3 and two clusters of size 1. Uniqueness and non uniqueness in percolation theory, probab. Pdf nonuniqueness and stability of the configuration of. More and more of the water droplets drifted t o the rod. Several uniqueness results, with references, can be found in.
Uniqueness of the infinite component for percolation on a. For iid percolation on z d, uniqueness of the infinite cluster is a classical result, while on certain other transitive graphs uniqueness. Uniqueness and non uniqueness in percolation theory. This paper is an uptodate introduction to the problem of. In case of nonuniqueness we say that there is a phase transition. Furthermore, the proof has certain consequences for the thermodynamic functions of percolation theory, such as the number of clusters per site and the connectivity functions. The most wellknown condition which implies uniqueness is dobrushins condition of weak dependence dobrushin 1968a. Numerous varied conjectures and problems are proposed, with the hope of setting goals for future research in percolation theory.
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