Backbone Exponent and Annulus Crossing Probability for 2d Percolation

In this talk I will first review the recent derivation of the backbone exponent for 2D percolation joint with Nolin, Qian, and Zhuang. The method is based on the coupling between SLE and Liouville quantum gravity (LQG), and the integrability of the Liouville conformal field theory (LCFT) that governs the LQG surfaces. The value of the backbone exponent turns out to be a transcendental number that is the root an elementary equation. Then I will present a forthcoming result joint with Zhuang that computes the exact probability that there are two disjoint paths of the same color crossing an annulus. The backbone exponent captures its leading asymptotic, while the rest of the roots in the aforementioned elementary equation captures the asymptotic of all the remaining terms. This derivation is based on the matter-Liouville-ghost decomposition of 2D quantum gravity coupled with conformal matter, which was understood in the annulus in a previous work joint with Ang and Remi. The same method also gives the probability that there is at least one path crossing the annulus, and the probability that there is at least one path of each color crossing the annulus, which were previously predicted by Cardy.