Zero-temperature Glauber dynamics on Z^d

We study zero-temperature Glauber dynamics on \Z^d, which is a dynamic version of the Ising model of ferromagnetism. Spins are initially chosen according to a Bernoulli distribution with density p, and then the states are continuously (and randomly) updated according to the majority rule. This corresponds to the sudden quenching of a ferromagnetic system at high temperature with an external field, to one at zero temperature with no external field. Define p_c(\Z^d) to be the infimum over p such that the system fixates at '+' with probability 1. It is a folklore conjecture that p_c(\Z^d) = 1/2 for every 2 \le d \in \N. We prove that p_c(\Z^d) \to 1/2 as d \to \infty.

[1]  P. Ney,et al.  Some Problems on Random Intervals and Annihilating Particles , 1974 .

[2]  Problèmes de récurrence concernant des mouvements aléatoires de particules sur Z avec destruction , 1977 .

[3]  D. Schwartz.  On Hitting Probabilities for an Annihilating Particle Model , 1978 .

[4]  P. Leath,et al.  Bootstrap percolation on a Bethe lattice , 1979 .

[5]  R. Arratia.  Site Recurrence for Annihilating Random Walks on $Z_d$ , 1983 .

[6]  Michael Aizenman,et al.  Metastability effects in bootstrap percolation , 1988 .

[7]  R. Schonmann.  On the Behavior of Some Cellular Automata Related to Bootstrap Percolation , 1992 .

[8]  Emilio N.M. Cirillo,et al.  Finite Size Scaling in Three-Dimensional Bootstrap Percolation , 1998 .

[9]  F. Martinelli.  Lectures on Glauber dynamics for discrete spin models , 1999 .

[10]  C. D. Howard,et al.  Zero-temperature ising spin dynamics on the homogeneous tree of degree three , 2000, Journal of Applied Probability.

[11]  D. L. Stein,et al.  Zero-temperature dynamics of Ising spin systems following a deep quench: results and open problems , 2000 .

[12]  Charles M. Newman,et al.  Dynamics of Ising spin systems at zero temperature , 2000 .

[13]  Clusters and recurrence in the two-dimensional zero-temperature stochastic ising model , 2001, math/0103050.

[14]  F. Manzo,et al.  The Threshold Regime of Finite Volume Bootstrap Percolation , 2001 .

[15]  Elchanan Mossel,et al.  Glauber dynamics on trees and hyperbolic graphs , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[16]  A. Holroyd.  Sharp metastability threshold for two-dimensional bootstrap percolation , 2002, math/0206132.

[17]  C. Wu.  Zero-Temperature Dynamics of Ising Models on the Triangular Lattice , 2002 .

[18]  Approach to Fixation for Zero-Temperature Stochastic Ising Models on the Hexagonal Lattice , 2001, math/0111170.

[19]  R. Schonmann,et al.  Stretched Exponential Fixation in Stochastic Ising Models at Zero Temperature , 2002 .

[20]  A. Sinclair,et al.  Glauber Dynamics on Trees: Boundary Conditions and Mixing Time , 2003, math/0307336.

[21]  C. D. Howard,et al.  The Percolation Transition for the Zero-Temperature Stochastic Ising Model on the Hexagonal Lattice , 2003 .

[22]  F. Martinelli,et al.  Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree , 2004, math/0412450.

[23]  Béla Bollobás,et al.  Bootstrap percolation on the hypercube , 2006 .

[24]  Yuval Peres,et al.  Bootstrap Percolation on Infinite Trees and Non-Amenable Groups , 2003, Combinatorics, Probability and Computing.

[25]  József Balogh,et al.  Bootstrap percolation on the random regular graph , 2007, Random Struct. Algorithms.

[26]  Svante Janson,et al.  On percolation in random graphs with given vertex degrees , 2008, 0804.1656.

[27]  R. Schonmann,et al.  Bootstrap Percolation on Homogeneous Trees Has 2 Phase Transitions , 2008 .

[28]  Béla Bollobás,et al.  Majority Bootstrap Percolation on the Hypercube , 2007, Combinatorics, Probability and Computing.

[29]  B. Bollob'as,et al.  Bootstrap percolation in three dimensions , 2008, 0806.4485.

[30]  M. Biskup,et al.  Metastable Behavior for Bootstrap Percolation on Regular Trees , 2009, 0904.3965.