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, 9:00am11:00am
Details
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 ncomponent \varphi ^4 models, but also the selfavoiding walk that has been observed to formally correspond to a "zerocomponent" 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 continuoustime weakly selfavoiding walk and of the ncomponent \varphi ^4 model on the 4dimensional lattice \bbbmath Z ^4. Although a "zerocomponent" 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 ncomponent \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 continuoustime weakly selfavoiding walk, we determine the decay of the critical generating function for the "watermelon" network consisting of p weakly mutually and selfavoiding 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.
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