• Publications
  • Influence
Probability and random processes
Events and their probabilities random variables and their distributions discrete random variables continuous random variables generating functions and their applications Markov chains convergence ofExpand
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The shortest-path problem for graphs with random arc-lengths
TLDR
We consider the problem of finding the shortest distance between all pairs of vertices in a complete digraph on n vertices, whose arc-lengths are non-negative random variables. Expand
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Influence and sharp-threshold theorems for monotonic measures
The influence theorem for product measures on the discrete space (0, 1} N may be extended to probability measures with the property of monotonicity (which is equivalent to "strong positiveExpand
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Cluster detection in networks using percolation
We consider the task of detecting a salient cluster in a sensor network, that is, an undirected graph with a random variable attached to each node. Motivated by recent research in environmentalExpand
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An upper bound for the number of spanning trees of a graph
  • G. Grimmett
  • Mathematics, Computer Science
  • Discret. Math.
  • 1 December 1976
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Strict inequality for critical values of Potts models and random-cluster processes
We prove that the critical value βc of a ferromagnetic Potts model is astrictly decreasing function of the strengths of interaction of the process. This is achieved in the (more) general context ofExpand
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Locality of connective constants
  • G. Grimmett, Zhongyang Li
  • Mathematics, Physics
  • Discret. Math.
  • 29 November 2014
TLDR
We prove a locality theorem for connective constants, namely, that the connective constant of two graphs are close in value whenever the graphs agree on a large ball around the origin (and a further condition is satisfied). Expand
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Entanglement in the Quantum Ising Model
We study the asymptotic scaling of the entanglement of a block of spins for the ground state of the one-dimensional quantum Ising model with transverse field. When the field is sufficiently strong,Expand
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Lipschitz percolation
We prove the existence of a (random) Lipschitz function F : Z → Z such that, for every x ∈ Z, the site (x, F (x)) is open in a site percolation process on Z. The Lipschitz constant may be taken to beExpand
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