Discrete Loewner evolution

TL;DR

This paper introduces a discrete version of Loewner evolution driven by a random walk, demonstrating its convergence to SLE, and explores its properties and phase transition behavior.

Contribution

It presents a novel discrete Loewner evolution model, proves its convergence to SLE, and analyzes its Markovian, symmetry, and phase transition properties.

Findings

Discrete Loewner evolution converges to SLE under rescaling

Discrete model exhibits Markovian and symmetry properties similar to SLE

Phase transition occurs at parameter value 4

Abstract

We study a one parameter family of discrete Loewner evolutions driven by a random walk on the real line. We show that it converges to the stochastic Loewner evolution (SLE) under rescaling. We show that the discrete Loewner evolution satisfies Markovian-type and symmetry properties analogous to SLE, and establish a phase transition property for the discrete Loewner evolution when the parameter equals 4.