Tightness of approximations to the chemical distance metric for simple conformal loop ensembles

Abstract

Suppose that $\Gamma$ is a conformal loop ensemble (CLE$_\kappa$) with simple loops ($\kappa \in (8/3,4)$) in a simply connected domain $D \subseteq {\mathbf C}$ whose boundary is itself a type of CLE$_\kappa$ loop. Let $\Upsilon$ be the carpet of $\Gamma$, i.e., the set of points in $D$ not surrounded by a loop of $\Gamma$. We prove that certain approximations to the chemical distance metric in $\Upsilon$ are tight. More precisely, for each path $\omega \colon [0,1] \to \Upsilon$ and $\epsilon > 0$ we let ${\mathfrak N}_\epsilon(\omega)$ be the Lebesgue measure of the $\epsilon$-neighborhood of $\omega$. For $z,w \in \Upsilon$ we let ${\mathfrak d}_\epsilon(z,w;\Gamma) = \inf_\omega {\mathfrak N}_\epsilon(\omega)$ where the infimum is over all paths $\omega \colon [0,1] \to \Upsilon$ with $\omega(0) = z$, $\omega(1) = w$ and let ${\mathfrak m}_\epsilon$ be the median of $\sup_{z,w \in \partial D} {\mathfrak d}_\epsilon(z,w;\Gamma)$. We prove that $(z,w) \mapsto {\mathfrak m}_\epsilon^{-1} {\mathfrak d}_\epsilon(z,w;\Gamma)$ is tight and that any subsequential limit defines a geodesic metric on $\Upsilon$ which is H\"older continuous with respect to the Euclidean metric. We conjecture that the subsequential limit is unique, conformally covariant, and describes the scaling limit of the chemical distance metric for discrete loop models which converge to CLE$_\kappa$ for $\kappa \in (8/3,4)$ such as the critical Ising model.