We investigate the existence of phase transitions for a class of continuum multi-type particle systems. The interactions act on hyperedges between the particles, allowing us to define a class of models with geometry-dependent interactions. We establish the existence of stationary Gibbsian point processes for this class of models. A phase transition is defined with respect to the existence of multiple Gibbs measures, and we establish the existence of phase transitions in our models by proving that multiple Gibbs measures exist. Our approach involves introducing a random-cluster representation for continuum particle systems with geometry-dependent interactions. We then argue that percolation in the random-cluster model corresponds to the existence of a phase transition. The originality in this research is defining a random-cluster representation for continuum models with hyperedge interactions, and applying this representation in order to show the existence of a phase transition. We mainly focus on models where the interaction is defined in terms of the Delaunay hypergraph. We find that phase transitions exist for a class of models where the interaction between particles is via Delaunay edges or Delaunay triangles.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:606137 |
| Date | January 2013 |
| Creators | Nollett, William R. |
| Publisher | University of Warwick |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://wrap.warwick.ac.uk/60465/ |
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