Backbone exponent and annulus crossing probability for planar percolation

TL;DR

This paper derives the backbone exponent for 2D percolation as a transcendental number and provides an exact formula for the probability of two disjoint crossing paths in an annulus, using SLE and LQG methods.

Contribution

It presents the first derivation of the backbone exponent as a root of an elementary equation and links it to conformal field theory via SLE and LQG techniques.

Findings

Backbone exponent is a transcendental number.

Exact formula for annulus crossing probability.

Backbone exponent related to roots of an elementary equation.

Abstract

We report the recent derivation of the backbone exponent for 2D percolation. In contrast to previously known exactly solved percolation exponents, the backbone exponent is a transcendental number, which is a root of an elementary equation. We also report an exact formula for the probability that there are two disjoint paths of the same color crossing an annulus. The backbone exponent captures the leading asymptotic, while the other roots of the elementary equation capture the asymptotic of the remaining terms. This suggests that the backbone exponent is part of a conformal field theory (CFT) whose bulk spectrum contains this set of roots. Our approach is based on the coupling between SLE curves and Liouville quantum gravity (LQG), and the integrability of Liouville CFT that governs the LQG surfaces.