The Arcsine law as the limit of the internal DLA cluster generated by Sinai's walk

TL;DR

This paper establishes that the limit of the internal DLA cluster generated by Sinai's walk converges to a functional of Brownian motion, providing a novel interpretation of the Arcsine law.

Contribution

It introduces a new interpretation of the Arcsine law as the limit of the internal DLA cluster driven by Sinai's walk, linking random walk models to Brownian functionals.

Findings

Limit of internal DLA cluster identified as a Brownian functional.

Provides a new interpretation of the Arcsine law.

Connects Sinai's walk with classical probability laws.

Abstract

We identify the limit of the internal DLA cluster generated by Sinai's walk as the law of a functional of a Brownian motion which turns out to be a new interpretation of the Arcsine law.