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

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;

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: 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]).

13 Sep 2012: K}. We aim to minimise the misclassification error rate orrisk :. Risk(C) = P(. C(X) 6= Y). September 13, 2012- 24. R.

20 Apr 2012: 24. (5.19). Hence the number of rounds before hitting the root has a Geometric distribution,so it has bounded expectation. ... 1. α2β(n1)1lk=2. Po(Zn, > 0,Zn,k > 0). (5.24). 25. We let An, = Jn,1 Jn, Jn,1, for = 1, ,α2β(n1) 1 and for = 0 we

21 Sep 2012: Γ(. 2 2d)2d/(d4). 24/(d4). R. J. Samworth Nonparametric classification. Further regret comparisons.

15 Aug 2012: 24 C. BEZUIDENHOUT, G. GRIMMETT, A. LOFFLERwhence, by Wald's equation ([13], p.