A recent paper due to Duminil-Copin and Tassion from 2019 introduces a novel
    argument for obtaining estimates on horizontal crossing probabilities of the
    random cluster model, in which a range of four possible behaviors is
    established. To apply the novel renormalization of crossing probabilities that
    the authors propose can be studied in other models of interest that are not
    self-dual, we collect results to formulate vertical and horizontal strip, and
    renormalization, inequalities for the dilute Potts model, whose measure is
    obtained from the high temperature expansion of the loop $O(n)$ measure
    supported over the hexagonal lattice in the presence of two external fields.
    The dilute Potts model was originally introduced in $1991$ by Nienhuis and is
    another model that enjoys the RSW box crossing property in the Continuous
    Critical phase, which is one of the four possible behaviors that the model is
    shown to enjoy. Through a combination of the Spatial Markov Property (SMP) and
    Comparison between Boundary Conditions (CBC) of the high-temperature spin
    measure, four phases of the dilute Potts model can be analyzed, exhibiting a
    class of boundary conditions upon which the probability of obtaining a
    horizontal crossing is significantly dependent. The exponential factor that is
    inserted into the Loop $O(n)$ model to quantify properties of the
    high-temperature phase is proportional to the summation over all spins, and the
    number of monochromatically colored triangles over a finite volume, which is in
    exact correspondence with the parameter of a Boltzmann weight introduced in
    Nienhuis’ 1991 paper detailing extensions of the $q$-state Potts model.
    Asymptotically, in the infinite volume limit we obtain strip and
    renormalization inequalities that provide conditions on the RSW constants $1-c$
    and $c$.

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