We consider a model for random loops on graphs which is inspired by the Feynman–Kac formula for the grand canonical partition function of an ideal gas. We associate to this model a corresponding occupation field, which is a positive random field detailing the total time spent by loops at each vertex. We argue that well known critical phenomena for the ideal gas can be reinterpreted in terms of random variables of this occupation field. We also argue that higher order correlations, such as the existence of off-diagonal long-range order, can only be seen in the occupation field by studying a modified space–time model of loops. We provide an isomorphism theorem for this model to a complex Gaussian field, and derive a version of Symanzik’s formula which describes the ideal gas interacting with a random background environment. Finally we consider the effect of interactions on the gas, and present a large deviations analysis of the cycle distribution of the loop model under two mean field Hamiltonians.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:694604 |
| Date | January 2015 |
| Creators | Daniel, Owen |
| Publisher | University of Warwick |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://wrap.warwick.ac.uk/81891/ |
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