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UBC Theses and Dissertations

Renormalization group analysis of self-interacting walks and spin systems Wallace, Benjamin


The central concern of this thesis is the study of critical behaviour in models of statistical physics in the upper-critical dimension. We study a generalized n-component lattice |φ|⁴ model and a model of weakly self-avoiding walk with nearest-neighbour contact self-attraction on the Euclidean lattice ℤd. By utilizing a supersymmetric integral representation involving boson and fermion fields, the two models are studied in a unified manner. Our main result, which is contingent on a small coupling hypothesis, identifies the precise leading-order asymptotics of the two-point function, susceptibility, and finite-order correlation length of both models in d = 4. In particular, we show that the critical two-point function satisfies mean-field scaling whereas the near-critical susceptibility and finite-order correlation length exhibit logarithmic corrections to mean-field behaviour. The proof employs a renormalization group method of Bauerschmidt, Brydges, and Slade based on a finite-range covariance decomposition and requires two extensions to this method. The first extension, which is required for the computation of the finite-order correlation length (even for the ordinary weakly self-avoiding walk and |φ|⁴ model), is an improvement of the norms used to control the evolution of the renormalization group. This allows us to obtain improved error estimates in the massive regime of the renormalization group flow. The second extension involves the identification of critical parameters for models initialized with a non-zero error coordinate coupled to a marginal/relevant coordinate. This allows us, for example, to realize the two-point function and susceptibility for the walk with self-attraction as a small perturbation of the corresponding quantities without self-attraction, whose asymptotic behaviour was determined by Bauerschmidt, Brydges, and Slade. This establishes a form of universality.

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