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Critical scaling limits and singular SPDES
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Gabriel, Simon (2023) Critical scaling limits and singular SPDES. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3984174
Abstract
In this thesis, we study scaling limits of idealised models arising in statistical physics. We present three works that, in different directions, explore critical behaviour of such systems.
In the first part, we study the 2D Allen–Cahn equation with white noise initial datum. This differential equation falls into the class of critical singular stochastic partial differential equations (SPDEs), for which no solution theory exists due to the roughness of the data. To give meaning to the equation, we consider the weak coupling scaling and establish non-trivial Gaussian fluctu- ations of the solution, by treating an infinite perturbative series expansion in terms of iterated stochastic integrals. To our best knowledge, this is the first time such an approach has been implemented for a (non-linear) critical SPDE and possibly a first step towards a general theory of SPDEs in this regime.
Closely connected, the second part of the thesis comprises the large-scale behaviour of the 2D directed random polymer model, describing the trajec- tory of a random walk in a random potential. In the weak disorder limit, we derive an invariance principle for the polymer paths. As a consequence, the random potential has no effect on large scales, which is due to a self- averaging effect. Similar results were previously only obtained for all but the (critical) two dimensional case.
Last, we study a system of particles with attractive interactions. Cluster- ing of particles occurs when tuning the system’s parameters, with growing system size. This describes a simplified model of (Bose–Einstein) condensa- tion. A careful analysis of the infinitesimal dynamics, using the Trotter–Kurtz approximation theorem, allows us to identify the limiting evolution in terms of a measure-valued Markov process. Moreover, we establish a link between the derived dynamics and the infinitely-many-neutral-alleles model in popu- lation genetics. The presented approach covers all interesting scaling regimes of the system parameters.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Limit theorems (Probability theory), Statistical mechanics, Stochastic partial differential equations, Polymers -- Mathematical models, Cluster theory (Nuclear physics) -- Mathematical models, Approximation theory, Markov processes | ||||
Official Date: | September 2023 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Zygouras, Nikos ; Chleboun, P. I. (Paul I.) | ||||
Sponsors: | University of Warwick. Mathematics Institute ; Engineering and Physical Sciences Research Council | ||||
Format of File: | |||||
Extent: | xii, 243 pages : illustrations | ||||
Language: | eng |
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