EXISTENCE OF A TRICRITICAL POINT FOR THE BLUME–CAPEL MODEL ON Z^d

We prove the existence of a tricritical point for the Blume–Capel model on Z^d for every d ≥ 2. The proof for d ≥ 3 relies on a novel combinatorial mapping to an Ising model on a larger graph, the techniques of Aizenman, Duminil-Copin, and Sidoravicious (Comm. Math. Phys. 334:2 (2015), 719–742), and the celebrated infrared bound. For d = 2, the proof relies on a quantitative analysis of crossing probabilities of the dilute random cluster representation of the Blume–Capel model. In particular, we develop a quadrichotomy result in the spirit of Duminil-Copin and Tassion (Moscow Math. J. 20:4 (2020), 711–740), which allows us to obtain a fine picture of the phase diagram for d = 2, including asymptotic behaviour of correlations in all regions. Finally, we show that the techniques used to establish subcritical sharpness for the dilute random cluster model extend to any d ≥ 2.