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26
Combinatorial approach to the interpolation method and scaling limits in sparse random graphs
, 2009
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Poisson process Fock space representation, chaos expansion and covariance inequalities
, 2009
"... We consider a Poisson process η on an arbitrary measurable space with an arbitrary sigmafinite intensity measure. We establish an explicit Fock space representation of square integrable functions of η. As a consequence we identify explicitly, in terms of iterated difference operators, the integrand ..."
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Cited by 29 (1 self)
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We consider a Poisson process η on an arbitrary measurable space with an arbitrary sigmafinite intensity measure. We establish an explicit Fock space representation of square integrable functions of η. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the WienerItô chaos expansion. We apply these results to extend wellknown variance inequalities for homogeneous Poisson processes on the line to the general Poisson case. The Poincaré inequality is a special case. Further applications are covariance identities for Poisson processes on (strictly) ordered spaces and HarrisFKGinequalities for monotone functions of η. Key words and phrases. Poisson process, chaos expansion, derivative operator, Kabanov
A sharper threshold for bootstrap percolation in two dimensions
"... Twodimensional bootstrap percolation is a cellular automaton in which sites become ‘infected ’ by contact with two or more already infected nearest neighbors. We consider these dynamics, which can be interpreted as a monotone version of the Ising model, on an n × n square, with sites initially inf ..."
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Cited by 14 (7 self)
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Twodimensional bootstrap percolation is a cellular automaton in which sites become ‘infected ’ by contact with two or more already infected nearest neighbors. We consider these dynamics, which can be interpreted as a monotone version of the Ising model, on an n × n square, with sites initially infected independently with probability p. The critical probability pc is the smallest p for which the probability that the entire square is eventually infected exceeds 1/2. Holroyd determined the sharp firstorder approximation: pc ∼ π2 /(18 log n) as n → ∞. Here we sharpen this result, proving that the second term in the expansion is −(log n) −3/2+o(1) , and moreover determining it up to a poly(log log n)factor.
ON THE SCALING LIMITS OF PLANAR PERCOLATION
, 2011
"... We prove Tsirelson’s conjecture that any scaling limit of the critical planar percolation is a black noise. Our theorems apply to a number of percolation models, including site percolation on the triangular grid and any subsequential scaling limit of bond percolation on the square grid. We also s ..."
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Cited by 13 (0 self)
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We prove Tsirelson’s conjecture that any scaling limit of the critical planar percolation is a black noise. Our theorems apply to a number of percolation models, including site percolation on the triangular grid and any subsequential scaling limit of bond percolation on the square grid. We also suggest a natural construction for the scaling limit of planar percolation, and more generally of any discrete planar model describing connectivity properties.
Lineofsight networks
 Proc. 18th ACMSIAM Symposium on Discrete Algorithms
, 2007
"... Random geometric graphs have been one of the fundamental models for reasoning about wireless networks: one places n points at random in a region of the plane (typically a square or circle), and then connects pairs of points by an edge if they are within a fixed distance of one another. In addition t ..."
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Cited by 10 (0 self)
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Random geometric graphs have been one of the fundamental models for reasoning about wireless networks: one places n points at random in a region of the plane (typically a square or circle), and then connects pairs of points by an edge if they are within a fixed distance of one another. In addition to giving rise to a range of basic theoretical questions, this class of random graphs has been a central analytical tool in the wireless networking community. For many of the primary applications of wireless networks, however, the underlying environment has a large number of obstacles, and communication can only take place among nodes when they are close in space and when they have lineofsight access to one another — consider, for example, urban settings or large indoor environments. In such domains, the standard model of random geometric graphs is not a good approximation of the true constraints, since it is not designed to capture the lineofsight restrictions. Here we propose a randomgraph model incorporating both range limitations and lineofsight constraints, and we prove asymptotically tight results for kconnectivity. Specifically, we consider points placed randomly on a grid (or torus), such that each node can see up to a fixed distance along the row and column it belongs to. (We think of the rows and columns as “streets ” and “avenues ” among a regularly spaced array of obstructions.) Further, we show that when the probability of node placement is a constant factor larger than the threshold for connectivity, nearshortest
LINEAR ALGEBRA AND BOOTSTRAP PERCOLATION
, 1107
"... Abstract. In Hbootstrap percolation, a set A ⊂ V(H) of initially ‘infected ’ vertices spreads by infecting vertices which are the only uninfected vertex in an edge of the hypergraph H. A particular case of this is the Hbootstrap process, in which H encodes copies of H in a graph G. We find the min ..."
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Cited by 6 (1 self)
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Abstract. In Hbootstrap percolation, a set A ⊂ V(H) of initially ‘infected ’ vertices spreads by infecting vertices which are the only uninfected vertex in an edge of the hypergraph H. A particular case of this is the Hbootstrap process, in which H encodes copies of H in a graph G. We find the minimum size of a set A that leads to complete infection when G and H are powers of complete graphs and H encodes induced copies of H in G. The proof uses linear algebra, a technique that is new in bootstrap percolation, although standard in the study of weakly saturated graphs, which are equivalent to (edge) Hbootstrap percolation on a complete graph. 1.
GRAPH BOOTSTRAP PERCOLATION
, 2011
"... Graph bootstrap percolation is a deterministic cellular automaton which was introduced by Bollobás in 1968, and is defined as follows. Given a graph H, and a set G ⊂ E(Kn) of initially ‘infected ’ edges, we infect, at each time step, a new edge e if there is a copy of H in Kn such that e is the on ..."
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Cited by 4 (0 self)
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Graph bootstrap percolation is a deterministic cellular automaton which was introduced by Bollobás in 1968, and is defined as follows. Given a graph H, and a set G ⊂ E(Kn) of initially ‘infected ’ edges, we infect, at each time step, a new edge e if there is a copy of H in Kn such that e is the only notyet infected edge of H. We say that G percolates in the Hbootstrap process if eventually every edge of Kn is infected. The extremal questions for this model, when H is the complete graph Kr, were solved (independently) by Alon, Kalai and Frankl almost thirty years ago. In this paper we study the random questions, and determine the critical probability pc(n,Kr) for the Krprocess up to a polylogarithmic factor. In the case r = 4 we prove a stronger result, and determine the threshold for pc(n,K4).
Percolation of arbitrary words in one dimension
, 2008
"... We consider a type of longrange percolation problem on the positive integers, motivated by earlier work of others on the appearance of (in)finite words within a site percolation model. The main issue is whether a given infinite binary word appears within an iid Bernoulli sequence at locations that ..."
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Cited by 4 (1 self)
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We consider a type of longrange percolation problem on the positive integers, motivated by earlier work of others on the appearance of (in)finite words within a site percolation model. The main issue is whether a given infinite binary word appears within an iid Bernoulli sequence at locations that satisfy certain constraints. We settle the issue in some cases, and provide partial results in others. 1
kclique Percolation and Clustering
, 2008
"... We summarise recent results connected to the concept of kclique percolation. This approach can be considered as a generalisation of edge percolation with a great potential as a community finding method in realworld graphs. We present a detailed study of the critical point for the appearance of a g ..."
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We summarise recent results connected to the concept of kclique percolation. This approach can be considered as a generalisation of edge percolation with a great potential as a community finding method in realworld graphs. We present a detailed study of the critical point for the appearance of a giant kclique percolation cluster in the Erdős–Rényigraph. The observed transition is continuous and at the transition point the scaling of the giant component with the number of vertices is highly nontrivial. The concept is extended to weighted and directed graphs as well. Finally, we demonstrate the effectiveness of kclique percolation as a community finding method via a series of realworld applications.