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Uniqueness and non-uniqueness in percolation theory
, 2005
"... Abstract: This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of infinite clusters for percolation on Z d and, more generally, on transitive graphs. For iid percolation on Z d, uniqueness of the infinite cluster is a classical result, while on certain other tr ..."
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Abstract: This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of infinite clusters for percolation on Z d and, more generally, on transitive graphs. For iid percolation on Z d, uniqueness of the infinite cluster is a classical result, while on certain other transitive graphs uniqueness may fail. Key properties of the graphs in this context turn out to be amenability and nonamenability. The same problem is considered for certain dependent percolation models – most prominently the Fortuin–Kasteleyn random-cluster model – and in situations where the standard connectivity notion is replaced by entanglement or rigidity. Socalled simultaneous uniqueness in couplings of percolation processes is also considered. Some of the main results are proved in detail, while for others the proofs are merely sketched, and for yet others they are omitted. Several open problems are discussed.
Limit Theorems for the Painting of Graphs By Clusters
- ESAIM PROBAB. STATIST
, 2001
"... We consider a generalization of the so-called divide and color model recently introduced by Häggström . We investigate the behaviour of the magnetization in large boxes and its fluctuations. Thus, laws of large numbers and Central Limit theorems are proved, both quenched and annealed. We show that t ..."
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Cited by 4 (2 self)
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We consider a generalization of the so-called divide and color model recently introduced by Häggström . We investigate the behaviour of the magnetization in large boxes and its fluctuations. Thus, laws of large numbers and Central Limit theorems are proved, both quenched and annealed. We show that the properties of the underlying percolation process roughly influence the behaviour of the colorying model. In the subcritical case, the limit magnetization is deterministic and the Central Limit Theorem admits a Gaussian limit. A contrario, the limit magnetization is not deterministic in the supercritical case and the limit of the Central Limit Theorem is not Gaussian, except in the particular model with exactly two colors which are equally probable.
Percolation in Correlated Systems
, 2007
"... In this thesis we study various problems in dependent percolation theory. In the first part of this thesis we study disordered q-state Potts models as examples of systems in which there is percolation for an arbitrary low density and no percolation for arbitrary high density of occupied sites. In th ..."
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Cited by 4 (0 self)
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In this thesis we study various problems in dependent percolation theory. In the first part of this thesis we study disordered q-state Potts models as examples of systems in which there is percolation for an arbitrary low density and no percolation for arbitrary high density of occupied sites. In the second part of the thesis we study dependent percolation models in which the correlations between the site occupation variables are long range, i.e. decaying as ra for a < d, where r is the separation between any two sites and d is the dimension of the model. Scaling analysis suggests [1] that such long range correlated percolation models dene a new percolation universality classes with critical exponents depending on a. We develop a field theoretic description of these models in an attempt to calculate the critical exponents of the transition in an double expansion in terms of = 6d and = 4a. In the third part we study the percolation transition for two specific long range correlated percolation models on the 3 dimensional integer square lattice. These two percolation models are derived from the Voter model and the Harmonic crystal respectively. Our simulation results conrm the basic scaling arguments and the field theoretic results.
Percolation Phenomena in Low and High Density Systems
"... that in the translation invariant measure, averaged over the disorder, at self–dual points any amalgamation of q − 1 species will fail to percolate despite an overall (high) density of 1 − q −1. Further, in the dilute bond version of these systems, if the system is just above threshold, then through ..."
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that in the translation invariant measure, averaged over the disorder, at self–dual points any amalgamation of q − 1 species will fail to percolate despite an overall (high) density of 1 − q −1. Further, in the dilute bond version of these systems, if the system is just above threshold, then throughout the low temperature phase there is percolation of a single species despite a correspondingly small density. Finally, we demonstrate both phenomena in a single model by considering a “perturbation ” of the dilute model that has a self–dual point. We also demonstrate that these phenomena occur, by a similar mechanism, in a simple coloring model invented by O. Häggström.
ELECTRONIC COMMUNICATIONS in PROBABILITY SOME TWO-DIMENSIONAL FINITE ENERGY PERCOLATION PRO-
, 2008
"... Some examples of translation invariant site percolation processes on the Z 2 lattice are constructed, the most far-reaching example being one that satisfies uniform finite energy (meaning that the probability that a site is open given the status of all others is bounded away from 0 and 1) and exhibi ..."
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Some examples of translation invariant site percolation processes on the Z 2 lattice are constructed, the most far-reaching example being one that satisfies uniform finite energy (meaning that the probability that a site is open given the status of all others is bounded away from 0 and 1) and exhibits a.s. the coexistence of an infinite open cluster and an infinite closed cluster. Essentially the same example shows that coexistence is possible between an infinite open cluster and an infinite closed cluster that are both robust under i.i.d. thinning. 1
CENTRAL LIMIT THEOREMS FOR THE POTTS MODEL
, 2005
"... We prove various q-dimensional Central Limit Theorems for the occurring of the colors in the q-state Potts model on Z d at inverse temperature β, provided that β is sufficiently far from the critical point βc. When (d = 2) and (q = 2 or q ≥ 26), the theorems apply for each β � = βc. In the uniquen ..."
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We prove various q-dimensional Central Limit Theorems for the occurring of the colors in the q-state Potts model on Z d at inverse temperature β, provided that β is sufficiently far from the critical point βc. When (d = 2) and (q = 2 or q ≥ 26), the theorems apply for each β � = βc. In the uniqueness region, a classical Gaussian limit is obtained. In the phase transition regime, the situation is more complex: when (q ≥ 3), the limit may be Gaussian or not, depending on the Gibbs measure which is considered. Particularly, we show that free boundary conditions lead to a non-Gaussian limit. Some particular properties of the Ising model are also discussed. The limits that are obtained are identified relatively to FK-percolation models.
