Asymmetry of near-critical percolation interfaces

TL;DR

This paper investigates the scaling limits of near-critical percolation interfaces on a triangular lattice, revealing that intermediate regimes produce limits singular to SLE(6) despite sharing the same Hausdorff dimension.

Contribution

It characterizes the properties of scaling limits in the intermediate near-critical regime, showing their singularity relative to SLE(6).

Findings

Scaling limits are singular to SLE(6) in the intermediate regime.

Large-scale behavior transitions from degenerate to SLE(6) as p(N) approaches 1/2.

All limits in the intermediate regime have Hausdorff dimension 7/4.

Abstract

We study the possible scaling limits of percolation interfaces in two dimensions on the triangular lattice. When one lets the percolation parameter p(N) vary with the size N of the box that one is considering, three possibilities arise in the large-scale limit. It is known that when p(N) does not converge to 1/2 fast enough, then the scaling limits are degenerate, whereas if p(N) - 1 / 2 goes to zero quickly, the scaling limits are SLE(6) as when p=1/2. We study some properties of the (non-void) intermediate regime where the large scale behavior is neither SLE(6) nor degenerate. We prove that in this case, the law of any scaling limit is singular with respect to that of SLE(6), even if it is still supported on the set of curves with Hausdorff dimension equal to 7/4.