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 $c$.
