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15 Aug 2012: Thus, from every pair Ci, Cj ofdistinct components, we obtain new elements F ′ of Fs1 by adding an edge joiningCi to Cj. ... their Applications 59 (1995), 267–275. 13. Janson, S., Luczak, T., Ruciński, A., Random Graphs, Wiley, New York, 2000.14.

15 Aug 2012: N ). We writeg = (gki : 1 i N ) for the associated function given by (2.12)–(2.14)subject to the new ordering, and C = g1(A). ... j1, j1,. , N}, and we construct a continuous increasing function g as abovebut subject to the new ordering.

15 Aug 2012: The principal new technique is aform of Russo's formula for random-cluster models reported by Bezuidenhout, Grimmett, andKesten.

15 Aug 2012: In this new network there are jeB(n)j parallelconnections between @B(n) and @B(n1), each of which has (for n M = M(Z))a resistance exceeding c= logn (by

2 Jul 2012: There has been a great deal of interest and activity in percolation theorysince the two Saint-Flour courses, [76, 111], of 1984 and 1996 reprinted inthis new edition. ... Press, New York, 1990,pp. 167–190. 91. O. Häggström, Uniqueness in

20 Dec 2012: We discuss these conditions in detail in Sections 3.2 and 3.3 respectively andcompare both the original and the new bounds to demonstrate the marked improvement. ... Our improved bounds are based on new versions of Markov’s inequality that hold for

15 Aug 2012: New difficulties arise through the presenceof orientations on the edges of the underlying lattice, and new ideas are neededto overcome these problems. ... 8, Springer, Berlin–Heidelberg–New York, 1987, pp. 13–20. 2. Aizenman, M., Newman, C.

15 Aug 2012: Having constructed tm1,we define tm = tm(X, X) by adding to tm1 a new edge 〈ξim , ξm〉 where 1 im < mand ξm ( X{ξ1, ξ2,. ,

15 Aug 2012: The percolation model contains a maxi-mum of (statistical) independence, and has proved a superb testing ground for new. ... Percolation theory has earned a reputation as a source of hard problems which areeasy to state and whose solutions require new

11 Apr 2012: U. In the latter case, we may remove U together with the previous DV stepto obtain a new admissible Λ-path from 0 to x of shorter length than π , a

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;