Conformal invariance of the loop-erased percolation explorer

TL;DR

This paper introduces a new loop-erased percolation interface model on the triangular lattice, demonstrating through simulations that its scaling limit is conformally invariant with a fractal dimension of 4/3, but not conforming to SLE_8/3.

Contribution

It defines a novel loop-erased percolation interface and provides evidence for its conformal invariance and fractal dimension via Monte Carlo simulations.

Findings

Scaling limit is conformally invariant

Fractal dimension is 4/3

Not described by SLE_8/3

Abstract

We consider critical percolation on the triangular lattice in a bounded simply connected domain with boundary conditions that force an interface between two prescribed boundary points. We say the interface forms a "near-loop" when it comes within one lattice spacing of itself. We define a new curve by erasing these near-loops as we traverse the interface. Our Monte Carlo simulations of this model lead us to conclude that the scaling limit of this loop-erased percolation interface is conformally invariant and has fractal dimension 4/3. However, it is not SLE_8/3. We also consider the process in which a near-loop is when the explorer comes within two lattice spacings of itself.