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15 Aug 2012: Rotator pin-ball. Ruijgrok and Cohen [24] have proposed a general studyof mirror and rotator' models. ... Quas, A., Some properties of Lorentz lattice gas models (1996) (to appear).24.

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

15 Aug 2012: min h, max h] =. [. 12,. 12. ]. (24). For a general unbiased walk, we take ascoin flip the unitary matrix. ... We have as beforethat the domain of the limit distribution is asin (24).

15 Aug 2012: mains unproved. Using Theorem 4.2 and an argument of Zhang (see [24, p. ... Related Fields 92, 511–527. 24. Grimmett, G. R. (1989), Percolation, Springer–Verlag, Berlin.

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

15 Aug 2012: Whenq = 2, this role is played by the Simon{Lieb inequality (see [24, 27]). ... 1986). Phase coexistence and surface tensions for the Pottsmodel. Communications in Mathematical Physics 105, 527{545.24.

15 Aug 2012: Mathematics and its Applications, vol. 99, Springer, New York, pp. 1–24.Aizenman, M.

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.

2 Jul 2012: Probab. 24 (1996), 1036–1048. 4. M. Atapour and N. Madras, On the number of entangled clusters, J.Statist. ... J. Probab. Stat. 24 (2010), 300–320. 114. H. Kesten, V. Sidoravicius, and Y.

15 Aug 2012: PERCOLATION ANDDISORDERED SYSTEMSGeorey GRIMMETT. 2PREFACEThis course aims to be a (nearly) self-contained account of part of the mathematicaltheory of percolation and related topics. The rst nine chapters summarise rigorousresults in percolation

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: Theorem 6. [24] We have that µp(K = 1) = 1 whenever p > pentc. ... Soc., 130:175–188, 2001. MR1797779. [24] O. Häggström. Uniqueness of the infinite entangled component in three-dimensional bond percolation.

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.

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: 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: 24 C. BEZUIDENHOUT, G. GRIMMETT, A. LOFFLERwhence, by Wald's equation ([13], p.

15 Aug 2012: CRITICAL PROBABILITIES FOR SITE. AND BOND PERCOLATION MODELS. G. R. Grimmett and A. M. Stacey. Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeAbstract. Any infinite graph G = (V, E) has a site percolation critical

15 Aug 2012: BRANCHING PROCESSES, AND. RANDOM-CLUSTER MEASURES ON TREES. Geoffrey Grimmett, Svante JansonAbstra t. Random-cluster measures on infinite regular trees are studied in con-junction with a general type of ‘boundary condition’, namely an

15 Aug 2012: THE RANDOM-CLUSTER MODEL. ON THE COMPLETE GRAPH. Béla Bollobás, Geoffrey Grimmett, Svante JansonAbstract. The random-cluster model of Fortuin and Kasteleyn contains as specialcases the percolation, Ising, and Potts models of statistical physics.

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: POTTS MODELS AND RANDOM-CLUSTER. PROCESSES WITH MANY-BODY INTERACTIONS. Geoffrey GrimmettAbstra t. Known differential inequalities for certain ferromagnetic Potts models with pair-interactions may be extended to Potts models with many-body

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

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

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.

15 Aug 2012: See [24, 51, 52] for more informa-tion and references.Few non-trivial facts are known about the limit set L, and much eort hasbeen spent, largely inconclusively, on attempting to

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.

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

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

15 Aug 2012: PERCOLATION ANDDISORDERED SYSTEMS. Georey GRIMMETT. 2PREFACEThis course aims to be a (nearly) self-contained account of part of the mathematicaltheory of percolation and related topics. The rst nine chapters summarise rigorousresults in percolation