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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: 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: PERCOLATION AND. DISORDERED SYSTEMS. Geoffrey GRIMMETT. Percolation and Disordered Systems 143. PREFACE. This course aims to be a (nearly) self-contained account of part of the math-ematical theory of percolation and related topics. The first nine

15 Aug 2012: 2.24) λ(C | Uj = 1) = λ(g(U ) A) λ(f (U ) A) = λ(B | Uj = 1).

15 Aug 2012: θΛ(p, q) = φΛ,p,q(A). (24). It is not hard to see that.

15 Aug 2012: the links between the Potts partition functionand the Tutte polynomial surveyed earlier by Welsh and Merino, [24]. ... 24] D. J. A. Welsh and C. Merino. The Potts model and the Tutte polyno-mial.

15 Aug 2012: 24] for a discussion). If d 3 and q is sufficiently large, then the uniqueness is aconsequence of Pirogov–Sinai theory ([37, 39]). ... 1992). Potts models and random-cluster processes with many-body interac-. tions (to appear).24.

15 Aug 2012: limΛLd. φ0Λ(A B) = Φ0(1A(ψ)µ0ψ(B)). (7.7). 24 B. T. Graham and G.

11 Apr 2012: Ax,m(n)) nnr=0 hmp (r). 1. (24). Once this is proved, it follows by (23) that. ... prove (24), and the proof is essentially that of [11], Lemma 5.17.

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