Three-dimensional tricritical spins and polymers

We consider two intimately related statistical mechanical problems on Z3: (i) the tricritical behavior of a model of classical unbounded n-component continuous spins with a triple-well single-spin potential (the |φ|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 |φ|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 renormalization group method that has been applied previously to analyze |φ|⁴ and weakly self-avoiding walk models on Z4.