A non-intersecting random walk on the Manhattan lattice and SLE_6

TL;DR

This paper investigates a constrained random walk on the Manhattan lattice, demonstrating through simulations that its scaling limit likely corresponds to SLE_6, linking lattice interfaces to conformal invariance in percolation.

Contribution

It provides empirical evidence supporting the conjecture that the scaling limit of the non-intersecting Manhattan lattice walk is SLE_6.

Findings

Strong support for the SLE_6 conjecture from Monte Carlo simulations.

The walk's properties align with theoretical predictions of SLE_6.

The model connects lattice percolation interfaces with conformal invariance principles.

Abstract

We consider a random walk on the Manhattan lattice. The walker must follow the orientations of the bonds in this lattice, and the walker is not allowed to visit a site more than once. When both possible steps are allowed, the walker chooses between them with equal probability. The walks generated by this model are known to be related to interfaces for bond percolation on a square lattice. So it is natural to conjecture that the scaling limit is SLE$_6$. We test this conjecture with Monte Carlo simulations of the random walk model and find strong support for the conjecture.