Coarsening Dynamics on Zd with Frozen Vertices

M. Damron; S. M. Eckner; H. Kogan; C. M. Newman; V. Sidoravicius

doi:10.1007/s10955-015-1247-4

Coarsening Dynamics on Z^d with Frozen Vertices

M. Damron, S. M. Eckner, H. Kogan, C. M. Newman, V. Sidoravicius

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We study Markov processes in which ±1-valued random variables σ_x(t),x∈Z^d, update by taking the value of a majority of their nearest neighbors or else tossing a fair coin in case of a tie. In the presence of a random environment of frozen plus (resp., minus) vertices with density ρ⁺ (resp., ρ^-), we study the prevalence of vertices that are (eventually) fixed plus or fixed minus or flippers (changing forever). Our main results are that, for ^ρ+>0 and ρ^-=0, all sites are fixed plus, while for ^ρ+>0 and ^ρ- very small (compared to ^ρ+), the fixed minus and flippers together do not percolate. We also obtain some results for deterministic placement of frozen vertices.

Original language	English (US)
Pages (from-to)	60-72
Number of pages	13
Journal	Journal of Statistical Physics
Volume	160
Issue number	1
DOIs	https://doi.org/10.1007/s10955-015-1247-4
State	Published - Jul 23 2015

Keywords

Coarsening models
Random environment
Zero-temperature Glauber dynamics

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

Access to Document

10.1007/s10955-015-1247-4

Cite this

@article{0bef1e7d4a674a32a986b03b6c68c47e,

title = "Coarsening Dynamics on Zd with Frozen Vertices",

abstract = "We study Markov processes in which ±1-valued random variables σx(t),x∈Zd, update by taking the value of a majority of their nearest neighbors or else tossing a fair coin in case of a tie. In the presence of a random environment of frozen plus (resp., minus) vertices with density ρ+ (resp., ρ-), we study the prevalence of vertices that are (eventually) fixed plus or fixed minus or flippers (changing forever). Our main results are that, for ρ+>0 and ρ-=0, all sites are fixed plus, while for ρ+>0 and ρ- very small (compared to ρ+), the fixed minus and flippers together do not percolate. We also obtain some results for deterministic placement of frozen vertices.",

keywords = "Coarsening models, Random environment, Zero-temperature Glauber dynamics",

author = "M. Damron and Eckner, {S. M.} and H. Kogan and Newman, {C. M.} and V. Sidoravicius",

note = "Funding Information: The authors thank Leo T. Rolla for many fruitful discussions. They also thank an anonymous referee for carefully reading the paper and suggesting several additional references. The research reported in this paper was supported in part by NSF Grants DMS-1007524 (S.E. and C.M.N.), DMS-1419230 (M.D.) and OISE-0730136 (S.E., H.K. and C.M.N.). V.S. was supported by ESF-RGLIS network and by Brazilian CNPq Grants 308787/2011-0 and 476756/2012-0 and FAPERJ Grant E-26/102.878/2012-BBP. Publisher Copyright: {\textcopyright} 2015, Springer Science+Business Media New York.",

year = "2015",

month = jul,

day = "23",

doi = "10.1007/s10955-015-1247-4",

language = "English (US)",

volume = "160",

pages = "60--72",

journal = "Journal of Statistical Physics",

issn = "0022-4715",

publisher = "Springer New York",

number = "1",

}

TY - JOUR

T1 - Coarsening Dynamics on Zd with Frozen Vertices

AU - Damron, M.

AU - Eckner, S. M.

AU - Kogan, H.

AU - Newman, C. M.

AU - Sidoravicius, V.

N1 - Funding Information: The authors thank Leo T. Rolla for many fruitful discussions. They also thank an anonymous referee for carefully reading the paper and suggesting several additional references. The research reported in this paper was supported in part by NSF Grants DMS-1007524 (S.E. and C.M.N.), DMS-1419230 (M.D.) and OISE-0730136 (S.E., H.K. and C.M.N.). V.S. was supported by ESF-RGLIS network and by Brazilian CNPq Grants 308787/2011-0 and 476756/2012-0 and FAPERJ Grant E-26/102.878/2012-BBP. Publisher Copyright: © 2015, Springer Science+Business Media New York.

PY - 2015/7/23

Y1 - 2015/7/23

N2 - We study Markov processes in which ±1-valued random variables σx(t),x∈Zd, update by taking the value of a majority of their nearest neighbors or else tossing a fair coin in case of a tie. In the presence of a random environment of frozen plus (resp., minus) vertices with density ρ+ (resp., ρ-), we study the prevalence of vertices that are (eventually) fixed plus or fixed minus or flippers (changing forever). Our main results are that, for ρ+>0 and ρ-=0, all sites are fixed plus, while for ρ+>0 and ρ- very small (compared to ρ+), the fixed minus and flippers together do not percolate. We also obtain some results for deterministic placement of frozen vertices.

AB - We study Markov processes in which ±1-valued random variables σx(t),x∈Zd, update by taking the value of a majority of their nearest neighbors or else tossing a fair coin in case of a tie. In the presence of a random environment of frozen plus (resp., minus) vertices with density ρ+ (resp., ρ-), we study the prevalence of vertices that are (eventually) fixed plus or fixed minus or flippers (changing forever). Our main results are that, for ρ+>0 and ρ-=0, all sites are fixed plus, while for ρ+>0 and ρ- very small (compared to ρ+), the fixed minus and flippers together do not percolate. We also obtain some results for deterministic placement of frozen vertices.

KW - Coarsening models

KW - Random environment

KW - Zero-temperature Glauber dynamics

UR - http://www.scopus.com/inward/record.url?scp=84931573482&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84931573482&partnerID=8YFLogxK

U2 - 10.1007/s10955-015-1247-4

DO - 10.1007/s10955-015-1247-4

M3 - Article

AN - SCOPUS:84931573482

VL - 160

SP - 60

EP - 72

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1

ER -