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Ising-Kac models near criticality

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Iberti, Massimo (2018) Ising-Kac models near criticality. PhD thesis, University of Warwick.

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Official URL: http://webcat.warwick.ac.uk/record=b3228143~S15

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Abstract

The present thesis consists in an investigation around the result shown by H. Weber and J.C. Mourrat in [MW17a], where the authors proved that the fluctuation of an Ising models with Kac interaction under a Glauber-type dynamic on a periodic two-dimensional discrete torus near criticality converge to the solution of the Stochastic Quantization Equation Φ 4/2.

In Chapter 2, starting from a conjecture in [SW16], we show the robustness of the method proving the convergence in law of the fluctuation field for a general class of ferromagnetic spin models with Kac interaction undergoing a Glauber dynamic near critical temperature. We show that the limiting law solves an SPDE that depends heavily on the state space of the spin system and, as a consequence of our method, we construct a spin system whose dynamical fluctuation field converges to Φ 2n/2.

In Chapter 3 we apply an idea by H. Weber and P. Tsatsoulis employed in [TW16], to show tightness for the sequence of magnetization fluctuation fields of the Ising-Kac model on a periodic two-dimensional discrete torus near criticality and characterise the law of the limit as the Φ 4/2 measure on the torus. This result is not an immediate consequence of [MW17a]. In Chapter 4 we study the fluctuations of the magnetization field of the Ising-Kac model under the Kawasaki dynamic at criticality in a one dimensional discrete torus, and we provide some evidence towards the convergence in law to the solution to the Stochastic Cahn-Hilliard equation.

Item Type: Thesis (PhD)
Subjects: Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH): Stochastic partial differential equations, Ising model, Ferromagnetism -- Mathematics, Magnetization -- Mathematical models, Besov spaces, Mathematical physics
Official Date: 2018
Dates:
DateEvent
2018Submitted
Institution: University of Warwick
Theses Department: Mathematics Institute
Thesis Type: PhD
Publication Status: Unpublished
Supervisor(s)/Advisor: Hairer, Martin
Format of File: pdf
Extent: v, 191 leaves
Language: eng

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