Singularity of Nearcritical Percolation Exploration Paths

TL;DR

This paper proves that the scaling limits of nearcritical percolation exploration paths with different parameters are mutually singular, meaning they are fundamentally different in distribution, and this difference can be detected from very small initial segments.

Contribution

It generalizes previous results by Nolin and Werner, showing singularity of nearcritical percolation exploration paths with different parameters and under scaling maps.

Findings

Scaling limits are mutually singular for different parameters.

Singularity can be detected from infinitesimal initial segments.

Nearcritical scaling limits are mutually singular under scaling maps.

Abstract

We show that the laws of scaling limits of nearcritical percolation exploration paths with different parameters are singular with respect to each other. This generalises a result of Nolin and Werner, using a similar technique. As a corollary, the singularity can even be detected from an infinitesimal initial segment. Moreover, nearcritical scaling limits of exploration paths are mutually singular under scaling maps.