Annulus crossing formulae for critical planar percolation

TL;DR

The paper provides rigorously derived exact annulus crossing formulae for critical planar percolation by embedding percolation in CLE$_6$ and coupling to Liouville quantum gravity on annuli. The authors obtain Cardy-type closed-channel and open-channel expressions for $p_B(\tau)$ and $p_{BW}(\tau)$ in terms of Dedekind eta functions, and derive a novel backbone-crossing formula $p_{BB}(\tau)$ with a backbone exponent defined by a transcendental equation, plus a spectrum of complex roots tied to subleading terms. The backbone analysis reveals logarithmic boundary-CFT structure and uses conformal welding of CLE/GFF-LQG quantum surfaces, along with KPZ-type relations, to extract the modulus distribution. The work provides a rigorous link between percolation observables and the geometry of random LQG surfaces, with the methodology extendable to FK-percolation, Potts, and O(n) loop models, and offers deeper insights into the CFT content of percolation through exact annulus observables. The results have implications for conformal field theory, probabilistic geometry, and the rigorous understanding of logarithmic behaviors in critical phenomena.

Abstract

We derive exact formulae for three basic annulus crossing events for the critical planar Bernoulli percolation in the continuum limit. The first is for the probability that there is an open path connecting the two boundaries of an annulus of inner radius $r$ and outer radius $R$. The second is for the probability that there are both open and closed paths connecting the two annulus boundaries. These two results were predicted by Cardy based on non-rigorous Coulomb gas arguments. Our third result gives the probability that there are two disjoint open paths connecting the two boundaries. Its leading asymptotic as $r/R\to 0$ is captured by the so-called backbone exponent, a transcendental number recently determined by Nolin, Qian and two of the authors. This exponent is the unique real root to the equation $\frac{\sqrt{36 x +3}}{4} + \sin (\frac{2 π\sqrt{12 x +1}}{3} ) =0$, other than $-\frac{1}{12}$ and $\frac{1}{4}$. Besides these three real roots, this equation has countably many complex roots. Our third result shows that these roots appear exactly as exponents of the subleading terms in the crossing formula. This suggests that the backbone exponent is part of a conformal field theory (CFT) whose bulk spectrum contains this set of roots. Expanding the same crossing probability as $r/R\to 1$, we obtain a series with logarithmic corrections at every order, suggesting that the backbone exponent is related to a logarithmic boundary CFT. Our proofs are based on the coupling between SLE curves and Liouville quantum gravity (LQG). The key is to encode the annulus crossing probabilities by the random moduli of certain LQG surfaces with annular topology, whose law can be extracted from the dependence of the LQG annuli partition function on their boundary lengths.

Figures (7)

• Figure 1: Illustration of $\widetilde{\mathcal{A}}_{BW}(A)$. The red boundaries represent $Q_i(\{0\} \times [0,1])$ and $Q_i(\{1 \} \times [0,1])$, while the blue boundaries represent $Q_i([0,1] \times \{0\})$ and $Q_i([0,1] \times \{1\})$ for $i \in \{1,2\}$. The red crossing in $Q_1$ represents an open crossing. The dashed blue crossing in $Q_2$ represents a closed crossing, which exists because $Q_2 \not \in \omega$.
• Figure 2: Left: Critical Bernoulli percolation on $\mathbb{D}_\delta$, where all vertices outside $\mathbb{D}_\delta$ (not drawn in the figure) are assumed to be black. The colored circuits are percolation interfaces between black and white clusters, with the blue circuit being the outermost one surrounding 0. As $\delta \rightarrow 0$, these circuits converge in law to CLE$_6$. Middle:$\mathcal{L}^o$ is the outermost CLE$_6$ loop that surrounds 0 and the light blue region $D_{\mathcal{L}^o}$ is the simply connected component of $\mathbb{D} \setminus \mathcal{L}^o$ that contains 0. In this figure, we have $\mathcal{L}^o \cap \partial \mathbb{D} \neq \emptyset$. Right: The blue loop $\mathcal{L}^*$ is the outermost CLE$_6$ loop whose outer boundary $\overline{\overline{\mathcal{L}^*}}$, colored in red, surrounds 0. The pink region $D_{\overline{\overline{\mathcal{L}^*}}}$ is the simply connected component of $\mathbb{D} \setminus \overline{\overline{\mathcal{L}^*}}$ that contains 0. In this figure, we have $\mathcal{L}^* \cap \partial \mathbb{D} = \emptyset$.
• Figure 3: Left:$\eta([0,\sigma_1])$ forms a counterclockwise loop, and the event $T$ occurs. $\mu^\circlearrowleft$ is the law of the loop formed by the solid blue and red curves, which is equivalent to the law of $\mathcal{L}^o$ restricted to the event $T$. Right:$\eta([0,\sigma_1])$ forms a clockwise loop, and the event $T$ does not occur. $\mu^\circlearrowright$ is the law of the loop formed by the solid blue and red curves.
• Figure 4: Left:$\eta([0, \sigma_1])$ forms a clockwise loop, and $\eta([\tau, \sigma_1])$ does not intersect $\partial \mathbb{D}$. In this case, the event $T^* \setminus T$ occurs. $\mu^*$ is the law of the loop formed by the solid blue and red curves, which is equivalent to the law of $\mathcal{L}^*$ restricted to the event $T^* \setminus T$. The measure $\mathsf{n}_{T^* \setminus T}$ is the law of the orange loop, which will be the welding interface in Lemma \ref{['lem:weld4.21']}. Right:$\eta([0, \sigma_1])$ forms a clockwise loop, and $\eta([\tau, \sigma_1])$ intersects $\partial \mathbb{D}$. In this case, the event $T^*$ does not occur. $\mu^{\circlearrowright,*}$ is the law of the loop formed by the solid blue and red curves, and $\mathsf{m}^*$ is the law of the orange loop, which will be the welding interface in Lemma \ref{['lem:weld4.23']}.
• Figure 5: Illustration of the events in Lemmas \ref{['lem:decorate-BA-1arm']} and \ref{['lem:decorate-BA-2arm']}. Sample $\phi$ from $\mathrm{LF}_\mathbb{H}^{(\gamma,i)}$ and an independent CLE$_6$ on $\mathbb{H}$. In both figures, we have $d_\phi(i ,\partial \mathbb{H}) > 1$. On this event, the pink region is $B^\bullet_{d_\phi} (i,1)$ whose quantum boundary length is $\mathscr{L}$, and the light blue region is $\mathcal{A}$. The blue loop $\mathcal{L}^i$ is the outermost CLE loop that surrounds $i$. Left: The event $F$ occurs, i.e., the pink region is surrounded by $\mathcal{L}^i$. Right: The event $G$ occurs, i.e., $\mathcal{L}^i$ touches the boundary of $\mathbb{H}$ and moreover the pink region is not surrounded by $\mathcal{L}^i$.

Theorems & Definitions (102)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Remark 2.2
• Lemma 2.3
• proof
• proof : Proof of Lemma \ref{['lem:percolation-limit']} for the case $\mathsf{a} \in \{ B, BW \}$
• Lemma 2.4
• proof
• Lemma 2.5