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15 Aug 2012: Each time we reach a new red vertex, there is astrictly positive probability that this vertex is the root of an ǫ-robust treein future generations. ... Soviet Mathe-matics Doklady 28, 23–26. Grimmett, G. R. (1989). Percolation. Springer-Verlag, New

15 Aug 2012: By(3.13), this process is supercritical. It follows that pc(k) p.(iv) A detailed argument would be long and contain no new ideas, and is thereforeomitted.

15 Aug 2012: families a new collection (Yx : x Zd) of random variables, which constitutes a site. ... another vertex is selected as a new starting point, and the process is iterated.

15 Aug 2012: We augment this graph by adding certainnew edges. Specifically, for distinct u,v Λ, we add a new edge between the pairu, v if either:(a) there exists a path of

15 Aug 2012: At each new point that it encounters, this point is a rw point withprobability prw, and the chance that r new points are not rw points is therefore (1 ... LK1. Now, by Theorem 3.4 and the remarks thereafter, each time thatthe above construction

15 Aug 2012: Thegeometry of the interfaces for this model is notably different from that of a spinmodel since the configurations are indexed by edges rather than by vertices, andthis leads to some new ... Although Dobrushin’s work is a helpfulindicator of the

15 Aug 2012: 6. Proofs of main results. In this section we combine the results derived so far with a new argument in orderto prove Theorems 2.1–2.3 for q 1. ... Inthe case q = 1, this gives new information about the probabilities of large deviationsof Cn.

15 Aug 2012: q} where q 1. We intro-duce this new model in Section 3, where we dub it the Blume–Capel–Potts(BCP) model.

15 Aug 2012: The only new complication is that for γF (0) > 1 the branching process.

15 Aug 2012: THE RANDOM-CLUSTER MODEL. Geoffrey GrimmettAbstra t. The class of random-cluster models is a unification of a variety of sto-chastic processes of significance for probability and statistical physics, including per-colation, Ising, and Potts models;

15 Aug 2012: The truth of such an assertionis sometimes far from evident. When studying such systems in generality, new difficultiesusually arise.

27 Jan 2012: Probability Surveys. Vol. 8 (2011) 294–332ISSN: 1549-5787DOI: 10.1214/11-PS185. Three theorems in. discrete random geometry. Geoffrey Grimmett. Centre for Mathematical SciencesUniversity of Cambridge. Wilberforce RoadCambridge CB3 0WB, UK. e-mail:

15 Aug 2012: the many open problems. Secondly, we present new results for such measures, as. ... model. In additionthis paper contains new results, as summarised later in this introduction.

15 Aug 2012: Their solutions require a mixture of new ideas, from analysis, geometry, anddiscrete mathematics.