The critical mean-field random cluster model
Lead Research Organisation:
UNIVERSITY OF OXFORD
Department Name: Statistics
Abstract
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Organisations
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509711/1 | 30/09/2016 | 29/09/2021 | |||
| 1796722 | Studentship | EP/N509711/1 | 30/09/2016 | 31/08/2020 | Alexander Homer |
| Description | The project is now complete. Though we were unable to prove that the critical mean-field random cluster model has a metric space scaling limit, as hoped, we did succeed in proving some preliminary results, and in finding a distribution which gives a plausible conjecture for the scaling limit. (A metric space scaling limit is, informally, a distribution which gives the approximate "shape" of the components of a random graph model, as the number of vertices tends to infinity and the length of the graph's edges is appropriately rescaled to keep the graph at "around the same size".) We also proved various results of independent interest, such as that the fixed-edge-count version of the Erdos-Renyi model has a scaling limit, and various facts about the number of surplus edges in the Erdos-Rényi model. (Surplus edges are edges in a graph additional to those necessary to form a forest whose components have the same vertex sets as the original graph.) |
| Exploitation Route | The thesis resulting from the project contains a variety of conjectures, proving an appropriate selection of which would lead to a full result. In addition, it may be possible to extend any result thus gained to a universality result: proving that the same limit occurs for a variety of related models. (Similar results exist for the Erdos-Rényi case.) |
| Sectors | Other |