Abstract

We prove the existence of a tricritical point for the Blume–Capel model on ${\mathbb{Z}}^d$ for every $d\ge 2$. The proof for $d\ge 3$ 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 $d\ge 2$.

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