Zero-temperature Glauber dynamics on ℤd

Abstract

We study zero-temperature Glauber dynamics on ℤd, 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 Pc(Zd) to be the infimum over p such that the system fixates at '+' with probability 1. It is a folklore conjecture that Pc(ℤd) = 1/2 for every 2 ≤ d ∈N. We prove that Pc(ℤd) → 1/2 as d → ∞. © 2009 Springer-Verlag.