Jonathan Tsai, Polytechnic University, Hong Kong

The scaling limit of 2-d myopic random walk

Abstract

Consider the hexagonal or square lattice in the two-dimensional upper half-plane. Myopic random walk is the following walk on the lattice from 0 to infinity: Starting from the origin, it moves to adjacent points chosen uniformly amongst the neighbouring points that do not lead to the walk getting trapped. It has been conjectured (by W. Werner and others) that the scaling limit (i.e. as the mesh size of the lattice tends to 0) of myopic random walk is SLE(6). In the hexagonal lattice, this is already known (Smirnov, Camia and Newman) due to its relation with percolation. We will present a proof of the convergence of myopic random walk to SLE(6) that does not require percolation. This is joint work with Phillip Yam.