Minkowski content construction of the CLE gasket measure
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866914478903263232 |
|---|---|
| author | Miller, Jason Yuan, Yizheng |
| author_facet | Miller, Jason Yuan, Yizheng |
| contents | We show for $κ\in (4,8)$ that the canonical conformally covariant measure on the conformal loop ensemble (CLE$_κ$) 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 $δ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13845 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Minkowski content construction of the CLE gasket measure Miller, Jason Yuan, Yizheng Probability Mathematical Physics Complex Variables 60G57, 60J67 (Primary) 28A80, 60F15, 60G18, 60K35 (Secondary) We show for $κ\in (4,8)$ that the canonical conformally covariant measure on the conformal loop ensemble (CLE$_κ$) 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 $δ$ 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. |
| title | Minkowski content construction of the CLE gasket measure |
| topic | Probability Mathematical Physics Complex Variables 60G57, 60J67 (Primary) 28A80, 60F15, 60G18, 60K35 (Secondary) |
| url | https://arxiv.org/abs/2604.13845 |