Critical Correlation Functions for the 4-Dimensional Weakly Self-Avoiding Walk and n-Component $${|\varphi|^4}$$ | φ | 4 Model

Abstract

We extend and apply a rigorous renormalisation group method to study critical correlation functions, on the 4-dimensional lattice Z 4 , for the weakly coupled n-component |ϕ| 4 spin model for all n ≥ 1, and for the continuous-time weakly self-avoiding walk. For the |ϕ| 4 model, we prove that the critical two-point function has |x| −2 (Gaussian) decay asymptotically, for n ≥ 1. We also determine the asymptotic decay of the critical correlations of the squares of components of ϕ, including the logarithmic corrections to Gaussian scaling, for n ≥ 1. The above 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 selfavoiding 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. In addition, for both models, we study the approach to the critical point and prove existence of logarithmic corrections to scaling for certain correlation functions. Our method gives a rigorous analysis of the weakly self-avoiding walk as the n = 0 case of the |ϕ| 4 model, and provides a unified treatment of both models, and of all the above results.

Three-dimensional tricritical spins and polymers

“…Much of our proof of Theorem 1.1 parallels the analysis used for the 4-dimensional case in [4,33]. In particular, with minor modifications, the results in [13,14] provide the analysis of a single RG step.…”

“…For dimension d = 4, the mean-field |x| −2 decay of the critical two-point function has been extensively studied [4,17,18,33], both for spin and polymer (weakly self-avoiding walk) models. On Z 4 , the mean-field behaviour results from infrared asymptotic freedom, which is itself closely connected to the marginal nature (in the RG sense) of |ϕ| 4 for d = 4.…”

“…The tuning of parameters follows as in [7] with only notational changes. The analysis of perturbation theory is also similar to that in [4,33], but here we present an improved and streamlined treatment of perturbation theory, which in particular demystifies the change of variables used in [6].…”

“…The calculation of the two-point function only requires the second derivative in (2.24), and therefore only needs the dependence on the observables to second order. As in [4,14,33], we formalise this simplification via use of a quotient space, in which two functions of ϕ, σ x , σ y become equivalent if their formal power series in the observable fields agree to order 1, σ x , σ y , σ x σ y . Thus we define N to consist of C 16 functions of ϕ of the form…”

“…For α = x and α = y, we make the scale dependent choice d 2 ½ j<jxy , where j xy is the coalescence scale. This is as in [4,33], after taking into account that the field dimension here is 1 2 rather than 1.…”