Three-dimensional tricritical spins and polymers

Abstract

We consider two intimately related statistical mechanical problems on ${\mathbb{Z}}^3$: (i) the tricritical behaviour of a model of classical unbounded $n$-component continuous spins with a triple-well single-spin potential (the $|\ddot{I}†{|}^6$ model), and (ii) a random walk model of linear polymers with a three-body repulsion and two-body attraction at the tricritical theta point (critical point for the collapse transition) where repulsion and attraction effectively cancel. The polymer model is exactly equivalent to a supersymmetric spin model which corresponds to the $n=0$ version of the $|\ddot{I}†{|}^6$ model. For the spin and polymer models, we identify the tricritical point, and prove that the tricritical two-point function has Gaussian long-distance decay, namely $|x{|}^{-1}$. The proof is based on an extension of a rigorous renormalisation group method that has been applied previously to analyse the $|\ddot{I}†{|}^4$ and weakly self-avoiding walk models on ${\mathbb{Z}}^4$.

Accepted version