Minkowski content construction of the CLE gasket measure

TL;DR

This paper demonstrates that the conformally covariant measure on the CLE gasket for rom 4 to 8 can be obtained as a limit of various approximation schemes, linking it to critical percolation cluster measures.

Contribution

It provides a new construction of the CLE gasket measure via Minkowski content and related methods, establishing finite moments of all orders.

Findings

The CLE gasket measure can be realized as a limit of Minkowski content and box-count variants.

The CLE6 gasket measure matches the conformally covariant measure from Garban-Pete-Schramm.

All fixed compact set measures have finite moments of all orders.

Abstract

We show for $\kappa \in (4,8)$ that the canonical conformally covariant measure on the conformal loop ensemble (CLE$_\kappa$) gasket, previously constructed indirectly by the first co-author and Schoug, can be realized as the limit of several natural approximation schemes. These include the Euclidean Minkowski content and its box-count variants, the properly renormalized number of dyadic squares that intersect the gasket, and the properly renormalized minimal number of balls of radius $\delta$ necessary to cover the gasket with respect to both its canonical geodesic and resistance metrics. This in particular allows us to identify the CLE$_6$ gasket measure with the conformally covariant measure constructed by Garban-Pete-Schramm as a scaling limit of the number of vertices in a macroscopic critical percolation cluster on the triangular lattice. Along the way, we show that the CLE gasket measure of every fixed compact set has finite moments of all orders; previously this was only known for first moments.