The critical mean-field random cluster model
Lead Research Organisation:
University of Oxford
Department Name: Statistics
Abstract
In a pair of papers in 2012 and 2010, Addario-Berry, Broutin and Goldschmidt showed the existence of, and gave the distribution for, the metric-space scaling limit of the Erdos-Rényi model in its critical regime, "critical" here referring to the critical point in the model's phase transition (a property first shown by Erdos and Rényi themselves). It is also known that several other random graph models possess a similar phase transition, with an analogous critical regime, such as the random cluster model, or various models of random planar graph. This project will explore a selection of such random graph models, with a view to proving similar scaling limit properties to those known for the Erdos-Rényi model.
Organisations
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509711/1 | 01/10/2016 | 30/09/2021 | |||
| 1796722 | Studentship | EP/N509711/1 | 01/10/2016 | 01/09/2020 | 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 |