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The random cluster model on finite graphs
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Mayes, Darion (2022) The random cluster model on finite graphs. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3821674
Abstract
The abrupt change of the size of the largest connected component is a central quantity of interest in the study of random graphs. For the percolation model, it is well known for a variety of families of finite regular graphs that the largest connected component experiences an asymptotic phase transition marking the emergence of a giant component (that is, one which contains a positive proportion of the total number of vertices) when the edge weight is appropriately rescaled by the vertex degree. For the random cluster model, a similar asymptotic phase transition was established by Bollobas, Grimmett, and Janson on the complete graph in [6]. The problem of establishing a similar phase transition on any family of finite graphs with more complicated geometry had remained open.
In this thesis, we study the emergence of the giant component for the random cluster model on two families of finite regular graphs. Our first result provides an alternative analysis of the random cluster model on the complete graph using a thermodynamic/ large deviations approach introduced by Biskup, Chayes, and Smith to study percolation on the complete graph in [3]. In particular, we compute the exponential rate of the large deviations of the size of the largest connected component of the random graph. Our second result establishes an asymptotic phase transition for the random cluster model on the hypercube when the cluster weight is an integer. In particular, we introduce a new concept which we call the sprinkled random cluster measure, which we combine with results obtained from an analysis of the asymptotics of a corresponding Potts model in order to extend the arguments of [1] to the random cluster model.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Random graphs, Phase transformations (Statistical physics) -- Mathematical models, Cluster analysis -- Mathematical models | ||||
Official Date: | February 2022 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Kotecký, R. (Roman) | ||||
Sponsors: | Engineering and Physical Sciences Research Council | ||||
Format of File: | |||||
Extent: | v, 107 leaves | ||||
Language: | eng |
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