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Hunting submartingales in the jumping voter model and the biased annihilating branching process

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Aidan Sudbury
Aidan Sudbury*
Affiliation:
Monash University
*
∗ Postal address: Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia. Email address: [email protected]
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Abstract

Two species (designated by 0's and 1's) compete for territory on a lattice according to the rules of a voter model, except that the 0's jump d0 spaces and the 1's jump d1 spaces. When d0 = d1 = 1 the model is the usual voter model. It is shown that in one dimension, if d1 > d0 and d0 = 1,2 and initially there are infinitely many blocks of 1's of length ≥ d1, then the 1's eliminate the 0's. It is believed this may be true whenever d1 > d0. In the biased annihilating branching process particles place offspring on empty neighbouring sites at rate λ and neighbouring pairs of particles coalesce at rate 1. In one dimension it is known to converge to the product measure density λ/(1+λ) when λ ≥ 1/3, and the initial configuration is non-zero and finite. This result is extended to λ ≥ 0.0347. Bounds on the edge-speed are given.

Keywords

Interacting particle systemsvoter modelsubmartingale

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 3 , September 1999 , pp. 839 - 854
Copyright
Copyright © Applied Probability Trust 1999 

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