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Clusters in middle-phase percolation on hyperbolic plane

Jan Czajkowski (2011)

Banach Center Publications

I consider p-Bernoulli bond percolation on transitive, nonamenable, planar graphs with one end and on their duals. It is known from [BS01] that in such a graph G we have three essential phases of percolation, i.e. 0 < p c ( G ) < p u ( G ) < 1 , where p c is the critical probability and p u -the unification probability. I prove that in the middle phase a.s. all the ends of all the infinite clusters have one-point boundaries in ∂ℍ². This result is similar to some results in [Lal].

Coexistence probability in the last passage percolation model is 6 - 8 log 2

David Coupier, Philippe Heinrich (2012)

Annales de l'I.H.P. Probabilités et statistiques

A competition model on ℕ 2 between three clusters and governed by directed last passage percolation is considered. We prove that coexistence, i.e. the three clusters are simultaneously unbounded, occurs with probability 6 - 8 log 2 . When this happens, we also prove that the central cluster almost surely has a positive density on ℕ 2 . Our results rely on three couplings, allowing to link the competition interfaces (which represent the borderlines between the clusters) to some particles in the multi-TASEP, and...

Cogrowth and spectral gap of generic groups

Yann Ollivier (2005)

Annales de l’institut Fourier

The cogrowth exponent of a group controls the random walk spectrum. We prove that for a generic group (in the density model) this exponent is arbitrarily close to that of a free group. Moreover, this exponent is stable under random quotients of torsion-free hyperbolic groups.

Collision probabilities in the rarefaction fan of asymmetric exclusion processes

Pablo A. Ferrari, Patricia Gonçalves, James B. Martin (2009)

Annales de l'I.H.P. Probabilités et statistiques

We consider the one-dimensional asymmetric simple exclusion process (ASEP) in which particles jump to the right at rate p∈(1/2, 1] and to the left at rate 1−p, interacting by exclusion. In the initial state there is a finite region such that to the left of this region all sites are occupied and to the right of it all sites are empty. Under this initial state, the hydrodynamical limit of the process converges to the rarefaction fan of the associated Burgers equation. In particular suppose that the...

Collisions of random walks

Martin T. Barlow, Yuval Peres, Perla Sousi (2012)

Annales de l'I.H.P. Probabilités et statistiques

A recurrent graph G has the infinite collision property if two independent random walks on G , started at the same point, collide infinitely often a.s. We give a simple criterion in terms of Green functions for a graph to have this property, and use it to prove that a critical Galton–Watson tree with finite variance conditioned to survive, the incipient infinite cluster in ℤ d with d ≥ 19 and the uniform spanning tree in ℤ 2 all have the infinite collision property. For power-law combs and spherically symmetric...

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