Four-Dimensional Weakly Self-avoiding Walk with Contact Self-attraction

Abstract

We consider the critical behaviour of the continuous-time weakly self-avoiding walk with contact self-attraction on Z 4 , for sufficiently small attraction. We prove that the susceptibility and correlation length of order p (for any p > 0) have logarithmic corrections to mean field scaling, and that the critical two-point function is asymptotic to a multiple of |x| −2 . This shows that small contact self-attraction results in the same critical behaviour as no contact self-attraction; a collapse transition is predicted for larger self-attraction. The proof uses a supersymmetric representation of the two-point function, and is based on a rigorous renormalisation group method that has been used to prove the same results for the weakly self-avoiding walk, without self-attraction. The model and main resultThe self-avoiding walk is a basic model for a linear polymer chain in a good solution. The repulsive self-avoidance constraint models the excluded volume effect of the polymer. In a poor solution, the polymer tends to avoid contact with the solution by instead making contact with itself. This is modelled by a self-attraction favouring nearest-neighbour contacts. The self-avoiding walk is already a notoriously difficult problem, and the combination of these two competing tendencies creates additional difficulties and an interesting phase diagram.In this paper, we consider a continuous-time version of the weakly self-avoiding walk with nearest-neighbour contact self-attraction on Z 4 . When both the self-avoidance and self-attraction are sufficiently weak, we prove that the susceptibility and finite-order correlation length have logarithmic corrections to mean field scaling with exponents 1 4 and 1 8 for the logarithm, respectively, and that the critical two-point function is asymptotic to a multiple of |x| −2 as |x| → ∞.

Mean-field tricritical polymers

“…In the mathematical literature, it has been more common to model the polymer collapse transition in terms of the interacting self-avoiding walk in which a walk with a self-repulsion receives an energetic reward for nearest-neighbour contacts. A review of the literature on this model can be found in [den Hollander 2009, Chapter 6]; more recent papers include [Bauerschmidt et al 2017;Hammond and Helmuth 2019;Pétrélis and Torri 2018]. In our mean-field model set on the complete graph, there is no geometry, and the notion of collapse (a highly localised walk) is less meaningful.…”

Three-dimensional tricritical spins and polymers

“…In [ 9 and ω 3 j (g 1/4 j ) 3 =g 3 j . This determined that we needed p N ≥ 9 + 1 = 10 derivatives of the field (see [14,Lemma 2.4]).…”

Untitled

“…In the mathematical literature, it has been more common to model the polymer collapse transition in terms of the interacting self-avoiding walk in which a walk with a self-repulsion receives an energetic reward for nearest-neighbour contacts. A review of the literature on this model can be found in [den Hollander 2009, Chapter 6]; more recent papers include [Bauerschmidt et al 2017;Hammond and Helmuth 2019;Pétrélis and Torri 2018]. In our mean-field model set on the complete graph, there is no geometry, and the notion of collapse (a highly localised walk) is less meaningful.…”