Existence and uniqueness of the conformally covariant geodesic metric on simple conformal loop ensemble carpets

TL;DR

This work constructs and proves the existence and uniqueness of a canonical geodesic metric on the CLE$_\kappa$ carpet for $\kappa\in(8/3,4)$, arising as the scaling limit of Minkowski first passage percolation. The authors develop a rigorous axiomatic framework and show that MFPP limits satisfy these axioms, which then determine the metric up to a global constant; they further establish a unique distance exponent $\theta(\kappa)$ governing scaling and relate it to Hausdorff dimensions of the carpet and geodesics. A key part of the approach mirrors the Liouville quantum gravity metric program, combining locality, translation/scale covariance, and tightness arguments to prove uniqueness. The results yield a robust geometric object for CLE carpets, with conjectured connections to scaling limits of discrete loop models and potential conformal covariance structures, and they lay groundwork for identifying the exact value of $\theta(\kappa)$ and extending to broader CLE regimes.

Abstract

We prove that for each $κ\in (8/3, 4)$ there exists a geodesic metric on the carpet of a CLE$_κ$ which is canonical in the sense that it is characterized by a certain list of axioms. Our metric can be constructed explicitly as the scaling limit of Minkowski first passage percolation (MFPP), i.e., the metric obtained by taking the infimum of the Lebesgue measure of the $\varepsilon$-neighborhood of all paths connecting each pair of points. Earlier work by the first co-author showed that MFPP admits nontrivial subsequential limits. The present paper shows that this subsequential limit is unique and is characterized by our list of axioms. We conjecture that our metric describes the scaling limit of the chemical distance metric for discrete loop models that converge to CLE$_κ$ for $κ\in (8/3, 4)$ in the scaling limit, e.g., the critical Ising model for $κ=3$. Our argument is inspired by recent works of Gwynne and Miller and Ding and Gwynne on the uniqueness of Liouville quantum gravity metrics.

Figures (12)

• Figure 1: Simulation of a metric ball associated with the $\mathrm{CLE}_\kappa$ metric with $\kappa \approx 2.72$ (top left), $\kappa=3$ (top right), $\kappa=10/3$ (bottom left), and $\kappa \approx 3.99$ (bottom right). Different colors represent the distance of points in the ball to the center. The $\kappa$ value associated with the top left simulation is close to the critical value $\kappa=8/3$ where the CLE carpet is the entire domain and the $\kappa$ value associated with the bottom right simulation is close to the critical value $\kappa=4$ above which the loops of a CLE intersect themselves, each other, and the domain boundary and the results of the present paper do not apply.
• Figure 2: Illustration of the proof of \ref{['214']}. The red path $P$ is contained in $B_{r_j}(z) \cap \Upsilon$. The times $\{t_n : n \in [1, N]_\mathbb Z\}$ are defined inductively so that for each $n \in [1, N - 1]_\mathbb Z$, there exists $k \in \mathbb N_{> j}$ such that $P(t_n) \in \partial B_{r_{j,k}}(z)$ and $t_{n + 1}$ is the first time after $t_n$ at which $P$ hits $\partial A_{r_{j,k - 1},r_{j,k + 1}}(z)$. The two blue loops are two loops of $\Gamma$ that cross between $\partial B_{r_{j,k}}(z)$ and $\partial B_{r_{j,k + 1}}(z)$. The set $\Upsilon_{k,l}$ is the connected component of $\overline{A_{r_{j,k},r_{j,k + 1}}(z)} \cap \Upsilon$ that lies between the two blue loops. We may assume without loss of generality that for each $\Upsilon_{k,l}$, there exists at most one $n \in [1, N]_\mathbb Z$ such that $P(t_n) \in \partial B_{r_{j,k}}(z) \cap \Upsilon_{k,l}$, and at most one $n \in [1, N]_\mathbb Z$ such that $P(t_n) \in \partial B_{r_{j,k + 1}}(z) \cap \Upsilon_{k,l}$. Then, since for each $k \in \mathbb N_{\ge j}$, we are able to bound the number of connected components of $\overline{A_{r_{j,k},r_{j,k + 1}}(z)} \cap \Upsilon$, and the $D_\Upsilon$-distance between any points that lie on the same such connected component, we are able to bound the $D_\Upsilon$-distance between $P(0)$ and $P(1)$.
• Figure 3: Illustration of the event $\overline H_r(M, a, \alpha)$ of \ref{['003']}. $P$ is a $D_\Upsilon$-geodesic in $\overline{A_{(1 - a)r,(1 + a)r}(0)}$ from $\partial B_{(1 - a)r}(0) \cap \Upsilon$ to $\partial B_{(1 + a)r}(0) \cap \Upsilon$. There are times $0 < s < t < 1$ such that $P(s) \in \partial B_{(1 - \alpha)r}(0) \cap \Upsilon$, $P(t) \in \partial B_r(0) \cap \Upsilon$, and $P|_{[s, t]} \subset \overline{A_{(1 - \alpha)r,r}(0)} \cap \Upsilon$. Also, $P|_{[s, t]}$ is a "shortcut", in the sense that $\widetilde{D}_\Upsilon(P(s), P(t)) \ge M D_\Upsilon(P(s), P(t))$.
• Figure 4: Illustration of the definition of the times $s_j$, $\sigma_j$, $\tau_j$, $t_j$ in the proof of \ref{['029']}. $t_j$ is the first time after $t_{j - 1}$ at which $P$ hits $\partial B_{(1 + a)r_j}(z_j)$; $s_j$ is the last time before $t_j$ at which $P$ hits $\partial B_{(1 - a)r_j}(z_j)$; $\sigma_j$ is the last time before $t_j$ at which $P$ hits $\partial B_{(1 - \alpha)r_j}(z_j)$; $\tau_j$ is the first time after $\sigma_j$ at which $P$ hits $\partial B_{r_j}(z_j)$.
• Figure 5: Illustration of the objects constructed in this section. The light green balls are the "test balls" that succeed, i.e., each of the light green balls contains a "good" pair of points $u$ and $v$, together with the $\widetilde{D}_\Upsilon$-geodesic connecting them (brown), and two connected arcs of $\Gamma$ (blue and solid). These connected arcs of $\Gamma$ are "rewired" in $\Gamma_r$ (blue and dashed) to form $\mathscr L_r$, $\gamma_r$, and $\gamma_r^\prime$. The red path $P_r$ is a $D_{\Upsilon_r}$-geodesic that passes though the "tube" between $\mathscr L_r$, $\gamma_r$, and $\gamma_r^\prime$. The times $\sigma$ and $\tau$ of \ref{['eq:093']} are so that $P_r(\sigma)$ is close to $u$ and $P_r(\tau)$ is close to $v$.

Theorems & Definitions (211)

• Lemma 1
• Definition 1
• Theorem 1.1: Convergence of MFPP
• Theorem 1.2: Uniqueness
• Theorem 1.3: Uniqueness of the distance exponent
• Theorem 1.4: Hausdorff dimension
• Definition 2
• Theorem 1.5
• Theorem 1.6
• Lemma 2