Boundary touching probability and nested-path exponent for non-simple CLE
The conformal loop ensemble (CLE) has two phases: for $\kappa \in (8/3, 4]$, the loops are simple and do not touch each other or the boundary; for $\kappa \in (4,8)$, the loops are non-simple and may touch each other and the boundary. For $\kappa\in(4,8)$, we derive the probability that the loop sur...
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Main Authors | , , , |
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Format | Journal Article |
Language | English |
Published |
29.01.2024
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Subjects | |
Online Access | Get full text |
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Summary: | The conformal loop ensemble (CLE) has two phases: for $\kappa \in (8/3, 4]$,
the loops are simple and do not touch each other or the boundary; for $\kappa
\in (4,8)$, the loops are non-simple and may touch each other and the boundary.
For $\kappa\in(4,8)$, we derive the probability that the loop surrounding a
given point touches the domain boundary. We also obtain the law of the
conformal radius of this loop seen from the given point conditioned on the loop
touching the boundary or not, refining a result of Schramm-Sheffield-Wilson
(2009). As an application, we exactly evaluate the CLE counterpart of the
nested-path exponent for the Fortuin-Kasteleyn (FK) random cluster model
recently introduced by Song-Tan-Zhang-Jacobsen-Nienhuis-Deng (2022). This
exponent describes the asymptotic behavior of the number of nested open paths
in the open cluster containing the origin when the cluster is large. For
Bernoulli percolation, which corresponds to $\kappa=6$, the exponent was
derived recently in Song-Jacobsen-Nienhuis-Sportiello-Deng (2023) by a color
switching argument. For $\kappa\neq 6$, and in particular for the FK-Ising
case, our formula appears to be new. Our derivation begins with Sheffield's
construction of CLE from which the quantities of interest can be expressed by
radial SLE. We solve the radial SLE problem using the coupling between SLE and
Liouville quantum gravity, along with the exact solvability of Liouville
conformal field theory. |
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DOI: | 10.48550/arxiv.2401.15904 |