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UBC Theses and Dissertations

Renormalisation group and critical correlation functions in dimension four Tomberg, Alexandre


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 |φ|⁴ models, but also the self-avoiding walk that has been observed to formally correspond to a "zero-component" spin model by de Gennes. Our subject in this thesis is the extension and application of a rigorous renormalisation group method developed by Bauerschmidt, Brydges and Slade to study the critical behaviour of the continuous-time weakly self-avoiding walk and of the n-component |φ|⁴ model on the 4-dimensional square lattice ℤ⁴. Although a "zero-component" vector is mathematically undefined (at least naively), we are able to interpret the weakly self-avoiding walk in a mathematically rigorous manner as the n = 0 case of the n-component |φ|⁴ model, and provide a unified treatment of both models. For the |φ|⁴ model, we determine the asymptotic decay of the critical correlation functions including the logarithmic corrections to Gaussian scaling, for n ≥ 1. This extends previously known results for n = 1 to all n ≥ 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 ≥ 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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