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 $d\geq 2$.

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