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15 Aug 2012: for all e lying in B(n). Once this is shown, we sum over e to obtain by (4.9)that. ... We claim that there exists an integer M, chosen uni-formly for edges e in B(n) and for all large n, such that.

11 Apr 2012: a) f (y) > 0, and(b) for all x Zd1, either f (x) = 0 or the site (x, f (x)) is open. ... Let d = 3. By (42), for all a (0, 12 ) there exists ζ > 0.

23 Jul 2012: cSpringer-Verlag 2006. 2 Random-Cluster Measures [1.1]. negative of the sum of σxσy over all edges e = 〈x, y〉 of G. ... Theintersection of the TF over all finite sets F is called the tail σ-field and is denotedby T.

15 Aug 2012: 4 J. van den Berg, G. R. Grimmett, R. B. Schinazi(In this case, coexistence occurs for all suciently large values of. ) ... factors are strictly positive and do not depend on A, it suces toprove that P;(B j A) is uniformly (over all events A of the form

2 Jul 2012: It is an openquestion to prove OZ decay for the truncated function all the way down tothe critical point. ... 7. J. Balogh, B. Bollobás, H. Duminil-Copin, and R. Morris, The sharpthreshold for bootstrap percolation in all dimensions, Trans.

15 Aug 2012: P(S A B) P(S A)P(S B). for all pairs A, B of increasing events with the property that there exists E′ Esuch that A is defined in ... It clearlysuffices to verify (5.1) for all E containing E1, respectively E2.

20 Dec 2012: 6/where D.η, t, B, r/ denotes the maximum of P.X t/ over all r-concave random variables sup-ported on {0, 1=B, 2=B,. , ... In all our simulations, we used B = 50.Each scenario was run 500 times and, to determine the set Lq=p, in each scenario, we

15 Aug 2012: We saythat A is fully connected to B if, for all b B, there exists a A such that a b;in this case, we write A fc B. ... Pp(E,′ ) 1 β(ǫ) for all , ′ D,. whenever M M(ǫ).Proof.

15 Aug 2012: By making all edges within B open, we may obtain thatPp(I m 1) > 0. ... PERCOLATION 15(b) L = Rd, and 1t fW(t) [M;M]d eventually, a.s.,for all M > 0.Case (b) holds if and only if a typical time coordinate T

15 Aug 2012: We. write = Zd = {0, 1}Z. d. , and C(R) = B(R), the set of all configurations on B(R). ... percolation. Then Yy = 0 and Yx = 1 for all x P1y P2y N.