Backbone exponent for two-dimensional percolation

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$_κ$ 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 $κ=6$ (corresponding to percolation), but for all $κ\in (4,8)$.

Figures (15)

• Figure 1: Site percolation on the triangular lattice is often represented as a random coloring of the faces of the dual hexagonal lattice. Left: The one-arm event corresponds to the existence of a black path connecting the vertex $0$ (indicated by a red hexagon) to distance $n$. Center: The four-arm event requests the existence of four paths with alternating colors, i.e. two black ones and two white ones, each connecting a neighbor of $0$ to distance $n$. It can be interpreted as the event that two distinct connected components "meet" in $0$. Right: In this paper, we determine the exponent corresponding to the existence of two disjoint black arms. This monochromatic two-arm exponent is most often called backbone exponent.
• Figure 2: This figure considers the (Euclidean) ball $B_n$ with radius $n=100$ centered on the vertex $0$ (indicated by a red hexagon). It shows the cluster of $0$ in $B_n$ at $p = p_c = \frac{1}{2}$, conditionally on the existence of one arm, that is, under the requirement that this cluster reaches the boundary of $B_n$. The backbone in the sense of Ke86b, i.e. the set of the vertices which are connected by two disjoint paths to 0 and the boundary, respectively, is then depicted in blue. In words, this means that the gray parts are those where a random walk is just "wasting time".
• Figure 3: In the left picture, we show an ${\rm SLE}_\kappa$ bubble $\eta$ on the complement of the event $E_i$. In the right picture, we show its outer boundary $\overline{\overline{\eta}}$, as well as the orange domain $D_i$ and the light cyan domain $D_{\overline{\overline{\eta}}}$. The domain $D_\infty$ refers to the union of the orange and pink regions.
• Figure 4: Let $0 \leq r_1 < r_2$. Given a percolation configuration in $B_{r_2}$, we consider interfaces, shown in red: they are self-avoiding paths, on the dual hexagonal lattice, separating black and white clusters. Left: If we consider black boundary conditions, which amounts to adding an extra layer of black sites (along $\partial^{\textrm{out}} B_{r_2}$), the set of interfaces is a collection of circuits. Center: There exists a white circuit surrounding the smaller ball $B_{r_1}$, i.e. there is no black arm in $A_{r_1,r_2}$. We consider the white connected component of this circuit, and its external edge boundary, which is a dual circuit. The external frontier is a black circuit, marked with crosses. Right: It there is a quasi-white circuit, containing exactly one black vertex, then there may exist one arm in $A_{r_1,r_2}$ (but not two disjoint arms). Again, we consider the white cluster of this circuit, its external edge boundary, and its external frontier.
• Figure 5: Black paths emanating from the external frontier of a loop may ensure the occurrence of the monochromatic two-arm event in the annulus $A_{r_1,r_2}$, through passages of width $1$, even in the case when that frontier surrounds $B_{r_1}$.

Theorems & Definitions (114)

• Theorem 1.1
• Theorem 1.2
• proof
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• proof
• Theorem 1.6
• Lemma 2.1
• Proposition 2.2