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Numerical study of Schramm-Loewner Evolution in the random 3-state Potts model

We have numerically studied the properties of the interface induced in the ferromagnetic random-bond three-state Potts model by symmetry-breaking boundary conditions. The fractal dimension $d_f$ of the interface was determined. The corresponding SLE parameter $κ$ was estimated to be $κ\simeq 3.18(6)$, compatible with previous estimate. On the other hand, we estimated $κ$ independently from the probability of passage of the interface at the left of a given point. The numerical data are well reproduced by the Schramm theoretical prediction and the fit leads to $κ\simeq 3.245(10)$, in agreement with the first estimate. This provides evidences that the geometric properties of spin interfaces in the random 3-state Potts model may be described by chordal ${\rm SLE}_κ$.