Overruled harmonic explorers in the plane and stochastic Löwner evolution

Abstract

The stochastic Löwner evolutions are a family of continuous stochastic curves in the real parameter , but only few and isolated examples of two-dimensional discrete growth processes areknown to converge to a . Although one-parameter families of discrete stochastic processes are provided by spin models such as theO(n) modelor the q-state Potts model, building an exploration process on the basis of these is onerous andcomputationally 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 ofcomputationally cheap exploration processes in one-to-one correspondence with . We introduce a class of exploration processes in the plane that extends the harmonicexplorer. These processes, dubbed overruled harmonic explorers, enjoy the domain Markovproperty by construction and are supposed to converge to in the scaling limit. We show by means of numerical simulations that crossingprobabilities in rectangular domains are indeed conformally invariant, and conjecture alinear relation between and the parameter p labelling the overruled harmonic explorer.