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Zero-temperature Glauber dynamics on \({\mathbb{Z}^d}\)
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  • Published: 11 December 2009

Zero-temperature Glauber dynamics on \({\mathbb{Z}^d}\)

  • Robert Morris1 

Probability Theory and Related Fields volume 149, pages 417–434 (2011)Cite this article

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Abstract

We study zero-temperature Glauber dynamics on \({\mathbb{Z}^d}\) , 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 \({p_c(\mathbb{Z}^d)}\) to be the infimum over p such that the system fixates at ‘ + ’ with probability 1. It is a folklore conjecture that \({p_c(\mathbb{Z}^d) = 1/2}\) for every \({2 \le d \in \mathbb{N}}\) . We prove that \({p_c(\mathbb{Z}^d) \to 1/2}\) as d → ∞.

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Authors and Affiliations

  1. Murray Edwards College, The University of Cambridge, Cambridge, CB3 0DF, England, UK

    Robert Morris

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  1. Robert Morris
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Correspondence to Robert Morris.

Additional information

The author was supported during this research by MCT grant PCI EV-8C. This work partly done whilst at the Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, Brazil.

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Morris, R. Zero-temperature Glauber dynamics on \({\mathbb{Z}^d}\) . Probab. Theory Relat. Fields 149, 417–434 (2011). https://doi.org/10.1007/s00440-009-0259-x

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  • Received: 19 February 2009

  • Revised: 22 November 2009

  • Published: 11 December 2009

  • Issue Date: April 2011

  • DOI: https://doi.org/10.1007/s00440-009-0259-x

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Mathematics Subject Classification (2000)

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