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Coarsening Dynamics on \(\mathbb {Z}^d\) with Frozen Vertices


We study Markov processes in which \(\pm 1\)-valued random variables \(\sigma _x(t), x\in \mathbb {Z}^d\), 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 \(\rho ^+\) (resp., \(\rho ^-\)), we study the prevalence of vertices that are (eventually) fixed plus or fixed minus or flippers (changing forever). Our main results are that, for \(\rho ^+ >0\) and \(\rho ^- =0\), all sites are fixed plus, while for \(\rho ^+ >0\) and \(\rho ^-\) very small (compared to \(\rho ^+\)), the fixed minus and flippers together do not percolate. We also obtain some results for deterministic placement of frozen vertices.

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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.

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Damron, M., Eckner, S.M., Kogan, H. et al. Coarsening Dynamics on \(\mathbb {Z}^d\) with Frozen Vertices. J Stat Phys 160, 60–72 (2015).

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  • Coarsening models
  • Zero-temperature Glauber dynamics
  • Random environment