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Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses
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In this article, the authors extend the theorem of Burton and Keane on uniqueness of the infinite component in dependent percolation to cover random graphs on Ω d or ℤ d × ℕ with long-range edges.Abstract:
We extend the theorem of Burton and Keane on uniqueness of the infinite component in dependent percolation to cover random graphs on ℤ d or ℤ d × ℕ with long-range edges. We also study a short-range percolation model related to nearest-neighbor spin glasses on ℤ d or on a slab ℤ d × {0,...K} and prove both that percolation occurs and that the infinite component is unique forV=ℤ2×{0,1} or larger.read more
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Probability on Trees and Networks
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TL;DR: In this article, the authors present a state-of-the-art account of probability on networks, including percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks.
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Processes on Unimodular Random Networks
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The Random-Cluster Model
TL;DR: The class of random-cluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures and of electrical networks as mentioned in this paper.
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Processes on Unimodular Random Networks
David Aldous,Russell Lyons +1 more
TL;DR: In this article, the authors investigate unimodular random networks and their properties via reversibility of an associated random walk and their similarities to unimmodular quasi-transitive graphs, and extend various theorems concerning random walks, percolation, spanning forests, and amenability.
References
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Journal ArticleDOI
Nonuniversal critical dynamics in Monte Carlo simulations
TL;DR: A new approach to Monte Carlo simulations is presented, giving a highly efficient method of simulation for large systems near criticality, despite the fact that the algorithm violates dynamic universality at second-order phase transitions.
Journal ArticleDOI
On the random-cluster model: I. Introduction and relation to other models
C.M. Fortuin,P.W. Kasteleyn +1 more
TL;DR: It is shown that the function which for the random-cluster model plays the role of a partition function, is a generalization of the dichromatic polynomial earlier introduced by Tutte, and related polynomials.
Journal ArticleDOI
A lower bound for the critical probability in a certain percolation process
TL;DR: In this paper, a random maze with the designations active or passive attached to the links is considered, where each link is assigned the designation active with probability p or passive with probability 1 − p, independently of all other links.
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