Mixing rate exponent of planar Fortuin-Kasteleyn percolation

Abstract

Duminil-Copin and Manolescu (2022) recently proved the scaling relations for planar Fortuin-Kasteleyn (FK) percolation. In particular, they showed that the one-arm exponent and the mixing rate exponent are sufficient to derive the other near-critical exponents. The scaling limit of critical FK percolation is conjectured to be a conformally invariant random collection of loops called the conformal loop ensemble (CLE). In this paper, we define the CLE analog of the mixing rate exponent. Assuming the convergence of FK percolation to CLE, we show that the mixing rate exponent for FK percolation agrees with that of CLE. We prove that the CLE$_κ$ mixing rate exponent equals $\frac{3 κ}{8}-1$, thereby answering Question 3 of Duminil-Copin and Manolescu (2022). The derivation of the CLE exponent is based on an exact formula for the Radon-Nikodym derivative between the marginal laws of the odd-level and even-level CLE loops, which is obtained from the coupling between Liouville quantum gravity and CLE.

Figures (4)

• Figure 1: Left: Primal and dual clusters are shown in gray and white, respectively. The event $A(r;\delta)$ requires a non-contractible dual circuit $\widetilde{\eta}$ (in dashed blue) surrounded by a non-contractible primal circuit $\eta$ (in red) in $\Lambda_{r,(1+\delta)r}$. Equivalently, the outer boundary $\mathcal{L}$ of a dual cluster lies in $\Lambda_{r,(1+\delta)r}$ and surrounds the origin, which should converge to a CLE loop of odd or even nesting level depending on the boundary conditions. Right: An illustration of the proof of Proposition \ref{['prop:boost-bc']}. We first reveal edges from outside in to find the outermost $\omega'$-open circuit $\eta'$ (in red) and $\omega$-open circuit $\eta$ (in blue) in $\Lambda_{r,(1+\delta)r}$. We partition $\Lambda_{(1+\frac{\delta}{3})r,(1+\frac{2\delta}{3})r}$ into $N$ annular tubes of width $\frac{\delta}{3N}$, then with positive probability $\eta'$ and $\eta$ are separated by one of these tubes (in gray). On the event $(\clubsuit) \cap (\spadesuit)$ (shown in green and orange), the event $A(r;\delta)$ occurs for $\omega'$ but not for $\omega$. (Figures are not to scale for illustrative reasons.)
• Figure 2: Construction of CLE$_\kappa$ via a branching SLE$_\kappa(\kappa-6)$ process. In this case $m=2$, the outermost loop $\mathcal{L}_1^0$ is the concatenation of the bold green and orange curves. By continuing $\eta$ beyond $\sigma_m$ towards the origin, we can construct a series of nested loops in the green region.
• Figure 3: Left: The graph of the $\frac{4}{\gamma^2}$-stable Lévy process $(X_t)_{t \ge 0}$ with only upward jumps. We draw a blue curve on the right of a red vertical line which corresponds to a jump, and identify the points that are on the same green horizontal line below the graph. Right: The $\frac{4}{\gamma^2}$-stable looptree of disks constructed from $(X_t)_{t \ge 0}$. The points on the red vertical line are collapsed to a single point. By replacing each topological disk with a quantum disk $\mathrm{QD}$ conditioned on having the same quantum boundary length as the size of the jump, we obtain a forested line. The quantum length of the line segment between the root $o$ and the point $p_t$ is $t$, and the generalized quantum length of the forested boundary between $o$ and $p_t$ is $Y_t = \inf\{s>0:X_s \le -t\}$.
• Figure 4: An illustration of Proposition \ref{['prop:single-loop-welding']}. Top: A forested pinched thin quantum annulus from $\mathsf{QA}_T^f$. Its conformal welding with the yellow forested disk from $\mathrm{QD}_{1,0}^f$ gives another sample from $\mathrm{QD}_{1,0}^f$ decorated with a boundary-touching CLE loop. The quantum surface $\mathcal{QA}_T$ considered in SXZ24 is the union of gray, purple and orange parts in the top-right panel, and is the conformal welding of $\mathsf{QA}_T$ and $\mathcal{M}^{\mathrm{circ}}$. Bottom: A forested quantum annulus from $\mathsf{QA}^f-\mathsf{QA}_T^f$. Its conformal welding with a forested disk from $\mathrm{QD}_{1,0}^f$ gives another sample from $\mathrm{QD}_{1,0}^f$ decorated with a non-boundary-touching CLE loop.

Theorems & Definitions (68)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Proposition 1.7
• Theorem 1.8
• proof : Proof of Theorem \ref{['thm:iota']} assuming Theorem \ref{['thm:iota-continuum-easy']} and Proposition \ref{['prop:event-A']}
• proof : Proof of Theorem \ref{['thm:iota-main']}
• Corollary 1.9