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 $mathbb{Z}^4$.
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