Backbone exponent and annulus crossing probability for planar percolation

TL;DR

The paper derives the planar percolation backbone exponent $x_B$ as the unique root in $(\tfrac{1}{4},\tfrac{2}{3})$ of $\frac{\sqrt{36 x +3}}{4} + \sin\left(\frac{2 \pi \sqrt{12 x +1}}{3}\right) = 0$, establishing the backbone dimension via $D_B = 2 - x_B$ and proving $x_B$ is transcendental. It provides an exact, q-series expression for the annulus crossing probability $p_B(r,R)$ in terms of spectral roots $s$ solving $\sin(4 \pi \sqrt{s/3}) + \tfrac{3}{2}\sqrt{s} = 0$ with $s = x + \tfrac{1}{12}$, thereby linking backbone observables to conformal field theory (CFT) data. The approach fuses the conformal radius encoding of SLE-boundary domains with 2D quantum gravity via Liouville CFT, exploiting the integrability of Liouville theory to derive exact partition-function identities and transform them into observables for percolation. The methodology extends to general CLE$_\kappa$ ($\kappa \in (4,8)$) and the $Q$-Potts universality class, suggesting a conformal or possibly logarithmic CFT interpretation of the backbone spectrum and offering a framework for exact exponents beyond previously rational values.

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.

Figures (4)

• Figure 1: Left: Bernoulli percolation on a triangular lattice on the unit disk. The blue loop is the outermost percolation interface surrounding the origin, whose scaling limit is $\eta_0$ on the right. Right: CLE$_6$ on the unit disk. The loops are nested, non-simple, and may touch each other and the boundary.
• Figure 2: Left: Percolation interfaces are colored orange. The blue loop is an example of a filled interface, which encloses all black points that can be surrounded by a white cluster together with an additional black point. It is the outermost one surrounding the origin. Right: Filled percolation interfaces converge to the outer boundaries of CLE$_6$ loops. The blue loop is the outermost one surrounding 0, and $D_b$ is the domain enclosed by this blue loop.
• Figure 3: Bernoulli site percolation on a random triangulation of the disk. The outermost filled percolation interface colored in blue divides the random triangulation into two parts. These two parts are independent conditioned on the number of vertices along the filled interface. In the continuum limit, this corresponds to \ref{['eq:key0']}.
• Figure 4: The two dashed lines represent two disjoint black crossings of $A(r, 1)$ on the event $d(0,\eta)>r$.