Zero-temperature Glauber dynamics on Z^d

2008/09/02 by Robert Morris, Morris, Robert · 1 citation
Mathematics · Physics and Astronomy · #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.0809.0353

openalex publication_date 2008/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 ≤ d ∈ \N. We prove that p_c(\Z^d) → 1/2 as d → ∞.

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