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Fluctuations of polymer partition functions and collision local times
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Lygkonis, Dimitrios (2022) Fluctuations of polymer partition functions and collision local times. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3884361
Abstract
In this work we study the directed polymer in random environment and some associated problems. In Chapter 2, we focus in spatial dimensions d ě 3 and study the spatial fluctuations of the field of partition functions and log-partition functions in the subregion of the weak disorder regime called L2 regime. We prove convergence of the two fields, under centering and suitable scaling, to the solution of the Edwards-Wilkinson model, thus establishing Gaussian fluctuations, in the full L2 regime. In Chapter 3 we study the directed polymer in random environment in the case of spatial dimension d “ 2 and in the so-called subcritical regime. We establish that all moments of the partition function are bounded in the full subcritical regime and compute their limit. As a byproduct, we obtain that the logarithmically scaled total collision local time between h, ph P N, h ě 3q, independent simple symmetric random walks on Z2 converges in distribution to a Gamma random variable. Based on this result, we formulate the conjecture that the joint distribution of the hph´1q{2 logarithmically scaled collision local times between h simple symmetric random walks on Z2 converges to that of a vector of hph ´ 1q{2 independent exponential random variables. Last, in Chapter 4, we prove the aforementioned conjecture on the logarithmically scaled collision local times by exactly computing their limiting joint Laplace transform. In order to prove this result, we build on tools developed in Chapter 3 and further analyse the microscopic structure of the collision local times.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics Q Science > QD Chemistry |
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Library of Congress Subject Headings (LCSH): | Polymers -- Statistical methods, Statistical mechanics, Gaussian processes, Partitions (Mathematics), Laplace transformation | ||||
Official Date: | 2022 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Zygouras, Nikos ; Cannizzaro, Giuseppe | ||||
Format of File: | |||||
Extent: | ix, 123 pages | ||||
Language: | eng |
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