Scaling limits of loop-erased Markov chains on resistance spaces via a partial loop-erasing procedure

TL;DR

This paper develops a framework for understanding the scaling limits of loop-erased Markov chains on resistance spaces, introducing partial loop-erasing operators and proving convergence to loop-erased paths on fractal structures.

Contribution

It introduces partial loop-erasing operators and demonstrates their use in constructing loop-erased paths on resistance spaces, including fractals like the Sierpiński carpet.

Findings

Loop-erased Markov chains converge to loop-erased paths on resistance spaces.

The constructed paths are simple and match the loop-erasure of diffusion paths.

Scaling limits of loop-erased random walks on Sierpiński carpet graphs are established.

Abstract

We introduce partial loop-erasing operators. We show that by applying a refinement sequence of partial loop-erasing operators to a finite Markov chain, we get a process equivalent to the chronological loop-erased Markov chain. As an application, we construct loop-erased random paths on bounded domains of resistance spaces as the weak limit of the loop erasure of the Markov chains on a sequence of finite sets approximating the space, and the limit is independent of the approximating sequences. The random paths we constructed are simple paths almost surely, and they can be viewed as the loop-erasure of the paths of the diffusion process. Finally, we show that the scaling limit of the loop-erased random walks on the Sierpi\'nski carpet graphs exists, and is equivalent to the loop-erased random paths on the Sierpi\'nksi carpet.