Backbone exponent for two-dimensional percolation

TL;DR

This paper derives an exact, transcendental value for the backbone exponent in 2D critical percolation using advanced mathematical tools like SLE, Liouville quantum gravity, and conformal field theory.

Contribution

It provides the first exact formula for the backbone exponent in 2D percolation, connecting it to SLE and Liouville quantum gravity.

Findings

Backbone exponent is a root of an elementary function.

The exponent has a transcendental value, unlike other rational arm exponents.

Derived a formula for all ppa rom 4 to 8.

Abstract

We derive an exact expression for the celebrated backbone exponent for Bernoulli percolation in dimension two at criticality. It turns out to be a root of an elementary function. Contrary to previously known arm exponents for this model, which are all rational, it has a transcendental value. Our derivation relies on the connection to the SLE$_\kappa$ bubble measure, the coupling between SLE and Liouville quantum gravity, and the integrability of Liouville conformal field theory. Along the way, we derive a formula not only for $\kappa=6$ (corresponding to percolation), but for all $\kappa \in (4,8)$.