Tid: 5 november 2001 kl 1515-1700
Plats : Seminarierummet 3733, Institutionen för matematik, KTH, Lindstedts väg 25, plan 7. Karta!
Föredragshållare: John Wierman, Johns Hopkins University, Baltimore
Titel: Critical Probabilities in Percolation Theory: Bounds, Conjectures, and Counterexamples.
Percolation models are random lattice graph models for critical phenomena. The critical probability, or percolation threshold, represents the phase transition point. Exact critical probability values are known for only a few two-dimensional lattices. Finding accurate bounds for critical probability values is a challenging problem.
The talk will describe the substitution method for deriving critical probability values. Improved bounds will be given for several lattices. The bounds are relevant to a conjecture of Häggström about vertex-transitive graphs, and lead to a conjecture regarding fully-triangulated lattices. Also, counterexamples are given to two common beliefs. They show that bond and site model critical probabilities need not be in the same order, and that the critical probability is not a monotone function of the average degree of the lattice. Implications for the physicists' development of universal formulas for predicting critical probabilities will be discussed.
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