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4,959
Sharp thresholds and percolation in the plane
, 2004
"... Recently, it was shown in [4] that the critical probability for random Voronoi percolation in the plane is 1/2. As a byproduct of the method, a short proof of the HarrisKesten Theorem was given in [5]. The aim of this paper is to show that the techniques used in these papers can be applied to many ..."
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Cited by 1 (0 self)
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Recently, it was shown in [4] that the critical probability for random Voronoi percolation in the plane is 1/2. As a byproduct of the method, a short proof of the HarrisKesten Theorem was given in [5]. The aim of this paper is to show that the techniques used in these papers can be applied
Exact Sampling with Coupled Markov Chains and Applications to Statistical Mechanics
, 1996
"... For many applications it is useful to sample from a finite set of objects in accordance with some particular distribution. One approach is to run an ergodic (i.e., irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has ..."
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Cited by 548 (13 self)
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For many applications it is useful to sample from a finite set of objects in accordance with some particular distribution. One approach is to run an ergodic (i.e., irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has run for M steps, with M sufficiently large, the distribution governing the state of the chain approximates the desired distribution. Unfortunately it can be difficult to determine how large M needs to be. We describe a simple variant of this method that determines on its own when to stop, and that outputs samples in exact accordance with the desired distribution. The method uses couplings, which have also played a role in other sampling schemes; however, rather than running the coupled chains from the present into the future, one runs from a distant point in the past up until the present, where the distance into the past that one needs to go is determined during the running of the al...
Community detection in graphs
, 2009
"... The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of th ..."
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Cited by 801 (1 self)
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The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such
The sharp threshold for bootstrap percolation in all dimensions
 In preparation
"... Abstract. In rneighbour bootstrap percolation on a graph G, a (typically random) set A of initially ‘infected ’ vertices spreads by infecting (at each time step) vertices with at least r alreadyinfected neighbours. This process may be viewed as a monotone version of the Glauber dynamics of the Isi ..."
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Cited by 39 (8 self)
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. In this paper we prove, for every pair d, r ∈ N with d � r � 2, that) d−r+1 d pc [n] , r = λ(d, r) + o(1) log (r−1)(n) as n → ∞, for some constant λ(d, r)> 0, and thus prove the existence of a sharp threshold for percolation in any (fixed) number of dimensions. We moreover determine λ(d, r) for every d � r
Topological estimation of percolation thresholds
 J. Stat. Mech. Theory Exp
, 2008
"... Abstract. Global physical properties of random media change qualitatively at a percolation threshold, where isolated clusters merge to form one infinite connected component. The precise knowledge of percolation thresholds is thus of paramount importance. For two dimensional lattice graphs, we use th ..."
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Cited by 5 (1 self)
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Abstract. Global physical properties of random media change qualitatively at a percolation threshold, where isolated clusters merge to form one infinite connected component. The precise knowledge of percolation thresholds is thus of paramount importance. For two dimensional lattice graphs, we use
Traffic and related selfdriven manyparticle systems
, 2000
"... Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ‘‘phantom traffic jams’ ’ even though drivers all like to drive fast? ..."
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Cited by 336 (38 self)
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Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ‘‘phantom traffic jams’ ’ even though drivers all like to drive fast? What are the mechanisms behind stopandgo traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ‘‘freeze by heating’’? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to selfdriven manyparticle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particlebased), mesoscopic (gaskinetic), and macroscopic (fluiddynamic) models. Attention is also paid to the formulation of a micromacro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for selfdriven manyparticle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socioeconomic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.
GroupInvariant Percolation on Graphs
 Geom. Funct. Anal
, 1999
"... . Let G be a closed group of automorphisms of a graph X . We relate geometric properties of G and X , such as amenability and unimodularity, to properties of Ginvariant percolation processes on X , such as the number of infinite components, the expected degree, and the topology of the components. O ..."
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Cited by 120 (40 self)
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. Let G be a closed group of automorphisms of a graph X . We relate geometric properties of G and X , such as amenability and unimodularity, to properties of Ginvariant percolation processes on X , such as the number of infinite components, the expected degree, and the topology of the components
Results 1  10
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4,959