We prove the existence of a tricritical point for the Blume-Capel model on
$\mathbb{Z}^d$ for every $d\geq 2$. The proof in $d\geq 3$ 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, 2015), and the
celebrated infrared bound. In $d=2$, the proof relies on a quantitative
analysis of crossing probabilities of the dilute random cluster representation
of the Blume-Capel. In particular, we develop a quadrichotomy result in the
spirit of Duminil-Copin and Tassion (Moscow Math. J., 2020), which allows us to
obtain a fine picture of the phase diagram in $d=2$, 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 $d\geq 2$.
