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The dilute Ising model : phase coexistence and slow dynamics in the region of phase transition

Abstract : This PhD thesis is concerned with the dilute Ising model, in the region of phase transition. The Ising model is a classical model of statistical mechanics; it has the peculiarity of having two distinct phases at low temperature, which motivated its use for the rigorous study of the phase coexistence phenomenon. Our objective was to extend the description of the phase coexistence phenomenon to the case of random media, that is to say, to the dilute Ising model, when the temperature and the dilution are weak enough for having two phases of opposite magnetization.

The thesis is made of four chapters. In a first chapter, we adapt the work of Pisztora to the random media and establish a coarse graining which is compatible with the dilution. In a second chapter, we study in detail the surface tension for that model, for both the Gibbs measure corresponding to a given realization of the media, and the averaged Gibbs measure. We characterize the low temperature limit of both quantities and describe the shape of the corresponding crystals. We show that lower deviations of surface tension happen at surface order and give a lower bound on the rate function with the help of concentration of measure theory. In a third chapter, we describe the phase coexistence phenomenon for both the Gibbs and averaged Gibbs measures. In a fourth and last chapter, we conclude the thesis with an application to the Glauber dynamics, and show that the autocorrelation decays not quicker than an inverse power of time.
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Submitted on : Saturday, April 12, 2008 - 10:56:22 PM
Last modification on : Wednesday, December 9, 2020 - 3:16:22 PM
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  • HAL Id : tel-00272899, version 1

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Marc Wouts. The dilute Ising model : phase coexistence and slow dynamics in the region of phase transition. Mathematics [math]. Université Paris-Diderot - Paris VII, 2007. English. ⟨tel-00272899⟩

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