Research output: Contribution to journal › Article

**Dependent Double Branching Annihilating Random Walk.** / Balazs, Marton; Nagy, Attila László.

Research output: Contribution to journal › Article

Balazs, M & Nagy, AL 2015, 'Dependent Double Branching Annihilating Random Walk', *Electronic Journal of Probability*, vol. 20, 84, pp. 1-32. https://doi.org/10.1214/EJP.v20-4045

Balazs, M., & Nagy, A. L. (2015). Dependent Double Branching Annihilating Random Walk. *Electronic Journal of Probability*, *20*, 1-32. [84]. https://doi.org/10.1214/EJP.v20-4045

Balazs M, Nagy AL. Dependent Double Branching Annihilating Random Walk. Electronic Journal of Probability. 2015 Aug 13;20:1-32. 84. https://doi.org/10.1214/EJP.v20-4045

@article{5c71c799dd2c42d8b86aad96d26ea39f,

title = "Dependent Double Branching Annihilating Random Walk",

abstract = "Double (or parity conserving) branching annihilating random walk, introduced by Sudbury in '90, is a one-dimensional non-attractive particle system in which positive and negative particles perform nearest neighbor hopping, produce two offsprings to neighboring lattice points and annihilate when they meet. Given an odd number of initial particles, positive recurrence as seen from the leftmost particle position was first proved by Belitsky, Ferrari, Menshikov and Popov in '01 and, subsequently in a much more general setup, in the article by Sturm and Swart (Tightness of voter model interfaces) in '08. These results assume that jump rates of the various moves do not depend on the configuration of the particles not involved in these moves. The present article deals with the case when the jump rates are affected by the locations of several particles in the system. Motivation for such models comes from non-attractive interacting particle systems with particle conservation. Under suitable assumptions we establish the existence of the process, and prove that the one-particle state is positive recurrent. We achieve this by arguments similar to those appeared in the previous article by Sturm and Swart. We also extend our results to some cases of long range jumps, when branching can also occur to non-neighboring sites. We outline and discuss several particular examples of models where our results apply.",

keywords = "non-attractive particle system, double branching annihilating random walk, parity conserving, long range dependent rates, long range branching, positive recurrence, interface tightness",

author = "Marton Balazs and Nagy, {Attila L{\'a}szl{\'o}}",

year = "2015",

month = "8",

day = "13",

doi = "10.1214/EJP.v20-4045",

language = "English",

volume = "20",

pages = "1--32",

journal = "Electronic Journal of Probability",

issn = "1083-6489",

publisher = "Institute of Mathematical Statistics",

}

TY - JOUR

T1 - Dependent Double Branching Annihilating Random Walk

AU - Balazs, Marton

AU - Nagy, Attila László

PY - 2015/8/13

Y1 - 2015/8/13

N2 - Double (or parity conserving) branching annihilating random walk, introduced by Sudbury in '90, is a one-dimensional non-attractive particle system in which positive and negative particles perform nearest neighbor hopping, produce two offsprings to neighboring lattice points and annihilate when they meet. Given an odd number of initial particles, positive recurrence as seen from the leftmost particle position was first proved by Belitsky, Ferrari, Menshikov and Popov in '01 and, subsequently in a much more general setup, in the article by Sturm and Swart (Tightness of voter model interfaces) in '08. These results assume that jump rates of the various moves do not depend on the configuration of the particles not involved in these moves. The present article deals with the case when the jump rates are affected by the locations of several particles in the system. Motivation for such models comes from non-attractive interacting particle systems with particle conservation. Under suitable assumptions we establish the existence of the process, and prove that the one-particle state is positive recurrent. We achieve this by arguments similar to those appeared in the previous article by Sturm and Swart. We also extend our results to some cases of long range jumps, when branching can also occur to non-neighboring sites. We outline and discuss several particular examples of models where our results apply.

AB - Double (or parity conserving) branching annihilating random walk, introduced by Sudbury in '90, is a one-dimensional non-attractive particle system in which positive and negative particles perform nearest neighbor hopping, produce two offsprings to neighboring lattice points and annihilate when they meet. Given an odd number of initial particles, positive recurrence as seen from the leftmost particle position was first proved by Belitsky, Ferrari, Menshikov and Popov in '01 and, subsequently in a much more general setup, in the article by Sturm and Swart (Tightness of voter model interfaces) in '08. These results assume that jump rates of the various moves do not depend on the configuration of the particles not involved in these moves. The present article deals with the case when the jump rates are affected by the locations of several particles in the system. Motivation for such models comes from non-attractive interacting particle systems with particle conservation. Under suitable assumptions we establish the existence of the process, and prove that the one-particle state is positive recurrent. We achieve this by arguments similar to those appeared in the previous article by Sturm and Swart. We also extend our results to some cases of long range jumps, when branching can also occur to non-neighboring sites. We outline and discuss several particular examples of models where our results apply.

KW - non-attractive particle system

KW - double branching annihilating random walk

KW - parity conserving

KW - long range dependent rates

KW - long range branching

KW - positive recurrence

KW - interface tightness

U2 - 10.1214/EJP.v20-4045

DO - 10.1214/EJP.v20-4045

M3 - Article

VL - 20

SP - 1

EP - 32

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

M1 - 84

ER -