# Three-dimensional tricritical spins and polymers

## TL;DR

This paper investigates the tricritical behavior of a three-dimensional spin model and an equivalent polymer model, establishing the decay of correlations at the tricritical point using advanced renormalisation group techniques.

## Contribution

It identifies the tricritical point in both models and proves Gaussian decay of the two-point function, extending renormalisation group methods to this new setting.

## Key findings

- Tricritical point identified in both models.
- Gaussian decay of the two-point function at tricriticality.
- Extension of renormalisation group methods to 3D tricritical models.

## 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 $|\varphi|^6$ 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 $|\varphi|^6$ 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 $|\varphi|^4$ and weakly self-avoiding walk models on $\mathbb{Z}^4$.