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PhD Candidate: Alexandre Tomberg
Mathematics, UBC
Thu 25 Jun 2015, 9:00am SPECIAL
One Time Event
Room 203, Graduate Student Centre, 6371 Crescent Rd., UBC
Doctoral Exam: Renormalisation Group and Critical Correlation Functions in Dimension Four
Room 203, Graduate Student Centre, 6371 Crescent Rd., UBC
Thu 25 Jun 2015, 9:00am-11:00am


Abstract: Critical phenomena and phase transitions are important subjects in statistical mechanics and probability theory. They are connected to the phenomenon of universality that makes the study of mathematically simple models physically relevant. Examples of such models include ferromagnetic spin systems such as the Ising, O(n) and n-component |\varphi |^4 models, but also the self-avoiding walk that has been observed to formally correspond to a "zero-component" spin model.

The subject of this thesis is the extension and application of a rigorous renormalisation group method developed by Brydges and Slade to study the critical behaviour of the continuous-time weakly self-avoiding walk and of the n-component |\varphi |^4 model on the 4-dimensional lattice \bbbmath Z ^4. Although a "zero-component" vector is mathematically undefined (at least naively), we are able to interpret the weakly selfavoiding walk in a mathematically rigorous manner as the n=0 case of the n-component |\varphi |^4 model, and provide a unified treatment of both models.

For the |\varphi |^4 model, we determine the asymptotic decay of the critical correlation functions including the logarithmic corrections to Gaussian scaling, for n\ge 1. This extends previously known results for n=1 to all n\ge 1, and also observes new phenomena for n>1, all with a new method of proof. For the continuous-time weakly self-avoiding walk, we determine the decay of the critical generating function for the "watermelon" network consisting of p weakly mutually and self-avoiding walks, for all p\ge 1, including the logarithmic corrections. This extends a previously known result for p=1, for which there is no logarithmic correction, to a much more general setting.