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23 Jul 2012: 1.24) η(ω) = {e E : ω(e) = 1}. Clearly,ω1 ω2 if and only if η(ω1) η(ω2).

15 Aug 2012: Our basic strategy in proving the central limit theorem isto adapt the arguments proposed by Kipnis and Varadhan [24] and further developed byDeMasi, Ferrari, Goldstein, and Wick [10, 11]. ... One of the main properties of the chain Xω is its

15 Aug 2012: Directed percolation is closely related to the contact model, for which blockarguments have been used to prove results related to some of those describedabove (see [7, 12, 23, 24]).

15 Aug 2012: Using (5.17)–(5.18) of[22], together with estimates at the beginning of the proof of Lemma (2.24) of [29],we find that. ... We have forf W(δ1) QG/3(e1) that. τ[QG/3(f) EL1,M1. ]= QG/3(τf) EL2,M2, (24).

15 Aug 2012: φbp,q = αφ′ (1 α)φ′′. for some distinct φ′,φ′′ Rp,q. It follows by [24, Thm. ... The given statement for θ0 may be proved similarly, making use of Theorem3.2 and [24, Prop.

20 Dec 2012: 104 5:12 104 1:32 103 2:59 103 4:37 1030.54 1:01 104 4:81 104 1:24 103 2:44 103 4:13 1030.55 ... 02 105 1:46 104 2:33 1040.88 7:64 106 3:12 105 7:24 105 1:32 104 2:11 1040.89 6:85 106 2:80 105

27 Jan 2012: gular and hexagonal lattices, [24],(c) the critical point of the random-cluster model on the square lattice with. ... 6See also [51]. Three theorems in discrete random geometry 315. Theorem 3.4 ([24]).

15 Aug 2012: F. (1986). Percolation. theory and some applications. Itogi Nauki i Techniki, Series of ProbabilityTheory, Mathematical Statistics, Theoretical Cybernetics, 24, 53–110.

15 Aug 2012: Et(ΨA) aΛs(A) b. for every increasing cylinder event A. Here are some remarks about these two lemmas, which are essentially equa-tions (13.24) and (13.25) of [7].

15 Aug 2012: 2.24) x = f (x). As is well known (see Harris (1963) proof of Theorem I.6.1) the only solutions of(2.24) in [0, 1] are q and 1. ... 24 GEOFFREY GRIMMETT AND HARRY KESTEN. as n. Proof. We prove.