Overruled harmonic explorers in the plane and stochastic Löwner evolution

The stochastic Löwner evolutions SLEκ are a family of continuous stochastic curves in the real parameter κ, but only few and isolated examples of two-dimensional discrete growth processes are known to converge to a SLEκ. Although one-parameter families of discrete stochastic processes are provided by spin models such as the O(n) model or the q-state Potts model, building an exploration process on the basis of these is onerous and computationally expensive, because it requires solving for the entire domain at each step. The basic idea of the present work is the search for a one-parameter family of computationally cheap exploration processes in one-to-one correspondence with SLEκ. We introduce a class of exploration processes in the plane that extends the harmonic explorer. These processes, dubbed overruled harmonic explorers, enjoy the domain Markov property by construction and are supposed to converge to SLEκ in the scaling limit. We show by means of numerical simulations that crossing probabilities in rectangular domains are indeed conformally invariant, and conjecture a linear relation between κ and the parameter p labelling the overruled harmonic explorer