Equivalence of Polychromatic Arm Probabilities on the Square Lattice

TL;DR

This paper proves an approximate color-switching lemma for multi-arm probabilities in 2D critical Bernoulli percolation on the square lattice, extending known results from triangular lattice and introducing a shifting transformation for primal-dual arm conversions.

Contribution

It introduces a new approximate color-switching lemma for polychromatic arm probabilities on the square lattice, addressing dual lattice complications with a novel shifting transformation.

Findings

Established an approximate color-switching lemma for square lattice percolation.

Extended known results from triangular to square lattice.

Developed a shifting transformation for primal-dual arm conversion.

Abstract

We consider 2d critical Bernoulli percolation on the square lattice. We prove an approximate color-switching lemma comparing k-arm probabilities for different polychromatic color sequences. This result is well-known for site percolation on the triangular lattice in [Nolin08]. To handle the complications arising from the dual lattice, we introduce a shifting transformation to convert arms between the primal and dual lattices.