| Summary: | <p>In this thesis we study the mean-field random cluster model, a random graph model in which, relative to the Erdős–Rényi model, graphs are biased to have a greater or lesser number of components, such that, in essence, large-deviations behaviour of the Erdős–Rényi model is seen. This model was independently proven to have an Erdős–Rényi-like phase transition by Stepanov [58, 59] and Bollobás, Grimmett and Janson [16], and was proven to have a critical window by Luczak and Łuczak [43]. We make progress towards demonstrating that the mean-field random cluster model also possesses a metric space scaling limit in its critical window, of the kind shown to exist for the Erdős–Rényi random graph by Addario-Berry, Broutin and Goldschmidt [3, 2].</p>
<p>We begin by reviewing results from the literature on scaling limits, and on limits of sequences of component sizes, for the Erdős–Rényi model; we also review existing results for the mean-field random cluster model. We go on to offer several results that form partial proofs of the existence of a scaling limit for the latter model, for differing values of the clustering parameter q, as well as some results which, while not proofs, are suggestive that the missing pieces can be filled. Along the way, we prove a number of results of independent interest about both models, including that the fixed-edge-count (as opposed to independent-edge) version of the Erdős–Rényi model has a scaling limit. We conclude by reviewing what would be necessary to complete the partial proofs, and make additional suggestions for future research.</p>
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