Possible Loss and Recovery of Gibbsianness¶During the Stochastic Evolution of Gibbs Measures |
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Authors: | ACD van Enter R Fernández F den Hollander F Redig |
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Affiliation: | (1) Instituut voor Theoretische Natuurkunde, Rijksuniversiteit Groningen, Nijenborg 4, 9747 AG Groningen, The Netherlands, NL;(2) Labo de Maths Raphael SALEM, UMR 6085, CNRS-Université de Rouen, Mathematiques, Site Colbert, 76821 Mont Saint Aignan, France, FR;(3) EURANDOM, Postbus 513, 5600 MB Eindhoven, The Netherlands, NL;(4) Faculteit Wiskunde en Informatica, Technische Universiteit Eindhoven, Postbus 513, 5600 MB Eindhoven, The Netherlands, NL |
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Abstract: | We consider Ising-spin systems starting from an initial Gibbs measure ν and evolving under a spin-flip dynamics towards a
reversible Gibbs measure μ≠ν. Both ν and μ are assumed to have a translation-invariant finite-range interaction. We study
the Gibbsian character of the measure νS(t) at time t and show the following:
(1) For all ν and μ, νS(t) is Gibbs for small t.
(2) If both ν and μ have a high or infinite temperature, then νS(t) is Gibbs for all t > 0.
(3) If ν has a low non-zero temperature and a zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t and non-Gibbs for large t.
(4) If ν has a low non-zero temperature and a non-zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t, non-Gibbs for intermediate t, and Gibbs for large t.
The regime where μ has a low or zero temperature and t is not small remains open. This regime presumably allows for many different scenarios.
Received: 26 April 2001 / Accepted: 10 October 2001 |
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