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

Ambrosio Valeria, University of Cambridge

In this talk we suppose that $\Gamma$ is a conformal loop ensemble (CLE_k) in the regime   $k\in (4, 8)$ where the loops intersect themselves and each others and sampled in a suitable simply connected domain $D\subseteq C$ whose boundary is itself de ned in terms of CLE_k loops. We let $\Upsilon$ be the gasket of $\Gamma$, i.e., the set of points in D not surrounded by a loop of $\Gamma$. We prove that a natural approximation procedure to the chemical distance metric in $\Upsilon$ is tight. We conjecture that the subsequential limit is unique, is conformally covariant, and describes the scaling limit of the chemical distance metric associated with discrete models which converge to such CLE_k's (e.g., critical percolation). Based on a joint work with Jason Miller and Yizheng Yuan.

Area: IS4 - Stochastic Geometry (Jacopo Borga)

Keywords: CLE

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