Displacement exponent for loop-erased random walk on the Sierpi\'nski gasket

TL;DR

This paper studies the behavior of loop-erased random walks on the Sierpinski gasket, establishing their asymptotic properties, including displacement exponent and law of the iterated logarithm, by extending finite walks to the infinite gasket.

Contribution

It introduces a method to extend finite loop-erased random walks to the infinite Sierpinski gasket and determines their asymptotic displacement behavior.

Findings

Displacement exponent for loop-erased random walk on Sierpinski gasket is established.

Law of the iterated logarithm for the walk is proved.

Extension method for finite to infinite gasket walks is developed.

Abstract

We prove that loop-erased random walks on finite pre-Sierpinski gaskets can be extended to the infinite pre-Sierpinski gasket by virtue of the `erasing-larger-loops-first' method, and obtain the asymptotic behavior of the walk as the number of steps increases, in particular, the displacement exponent and a law of the iterated logarithm.