Connection Probabilities for Conformal Loop Ensembles

@article{Miller2017ConnectionPF,
title={Connection Probabilities for Conformal Loop Ensembles},
author={Jason Miller and Wendelin Werner},
journal={Communications in Mathematical Physics},
year={2017},
volume={362},
pages={415-453}
}
• Published 9 February 2017
• Mathematics
• Communications in Mathematical Physics
The goal of the present paper is to explain, based on properties of the conformal loop ensembles $${{\rm CLE}_\kappa}$$CLEκ (both with simple and non-simple loops, i.e., for the whole range $${\kappa \in (8/3, 8)}$$κ∈(8/3,8)), how to derive the connection probabilities in domains with four marked boundary points for a conditioned version of $${{\rm CLE}_\kappa}$$CLEκ which can be interpreted as a $${{\rm CLE}_\kappa}$$CLEκ with wired/free/wired/free boundary conditions on four boundary arcs…
• Mathematics
Probability Theory and Related Fields
• 2021
We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble $$\hbox {CLE}_{\kappa '}$$ CLE κ ′ for $$\kappa '$$ κ ′ in (4, 8)
• Mathematics
Journal of the European Mathematical Society
• 2019
We show that when observing the range of a chordal SLE$_\kappa$ curve for $\kappa \in (4,8)$, it is not possible to recover the order in which the points have been visited. We also derive related
• Mathematics
Forum of Mathematics, Pi
• 2017
Conformal loop ensembles (CLEs) are random collections of loops in a simply connected domain, whose laws are characterized by a natural conformal invariance property. The set of points not surrounded
• Dapeng Zhan
• Mathematics
Communications in Mathematical Physics
• 2020
A 2- $$\hbox {SLE}_\kappa$$ SLE κ ( $$\kappa \in (0,8)$$ κ ∈ ( 0 , 8 ) ) is a pair of random curves $$(\eta _1,\eta _2)$$ ( η 1 , η 2 ) in a simply connected domain D connecting two pairs of
• Physics
Forum of Mathematics, Pi
• 2022
Abstract This paper studies the critical and near-critical regimes of the planar random-cluster model on $\mathbb Z^2$ with cluster-weight $q\in [1,4]$ using novel coupling techniques. More
• Hao Wu
• Mathematics
Communications in Mathematical Physics
• 2020
This article pertains to the classification of pairs of simple random curves with conformal Markov property and symmetry. We give the complete classification of such curves: conformal Markov property
• Mathematics
• 2019
We apply the Grassmannian representation of the dimer model, an equivalent approach to Kasteleyn's solution to the close-packed dimer problem, to calculate the connection probabilities for the
• Hao Wu
• Materials Science
Communications in Mathematical Physics
• 2020
This article pertains to the classification of pairs of simple random curves with conformal Markov property and symmetry. We give the complete classification of such curves: conformal Markov property
This article pertains to the classification of pairs of simple random curves with conformal Markov property and certain symmetry. Such pairs correspond to scaling limits of pairs of interfaces in
• Dapeng Zhan
• Materials Science
Communications in Mathematical Physics
• 2020
A 2-SLEκ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}

References

SHOWING 1-10 OF 65 REFERENCES

• Mathematics
Journal of the European Mathematical Society
• 2019
We show that when observing the range of a chordal SLE$_\kappa$ curve for $\kappa \in (4,8)$, it is not possible to recover the order in which the points have been visited. We also derive related
• Mathematics
• 2006
AbstractThe conformal loop ensembles CLEκ, defined for 8/3 ≤ κ ≤ 8, are random collections of loops in a planar domain which are conjectured scaling limits of the O(n) loop models. We calculate the
• Mathematics
Forum of Mathematics, Pi
• 2017
Conformal loop ensembles (CLEs) are random collections of loops in a simply connected domain, whose laws are characterized by a natural conformal invariance property. The set of points not surrounded
• Physics
• 2014
There is a simple way to "glue together" a coupled pair of continuum random trees (CRTs) to produce a topological sphere. The sphere comes equipped with a measure and a space-filling curve (which
Schramm-Loewner Evolutions ($\SLE$) are random curves in planar simply connected domains; the massless (Euclidean) free field in such a domain is a random distribution. Both have conformal invariance
• Mathematics
• 2013
We compute the almost-sure Hausdorff dimension of the double points of chordal $$\mathrm {SLE}_\kappa$$SLEκ for $$\kappa > 4$$κ>4, confirming a prediction of Duplantier–Saleur (1989) for the
• Mathematics
• 2012
Suppose that D is a planar Jordan domain and x and y are distinct boundary points of D. Fix \kappa \in (4,8) and let \eta\ be an SLE_\kappa process from x to y in D. We prove that the law of the
• Mathematics
Communications in mathematical physics
• 2019
The loop ensemble contains unboundedly many loops and hence the result describes the joint law of infinitely many loops in terms of SLE type processes, and the result gives the full scaling limit of the FK Ising model in the sense of random geometry of the interfaces.
We consider the planar Ising model in rectangle $(\Omega; x^L, x^R, y^R, y^L)$ with alternating boundary condition: $\ominus$ along $(x^Lx^R)$ and $(y^Ry^L)$, $\xi^R\in\{\oplus, \text{free}\}$ along
• Mathematics
Journal of the Institute of Mathematics of Jussieu
• 2017
We study certain classes of local sets of the two-dimensional Gaussian free field (GFF) in a simply connected domain, and their relation to the conformal loop ensemble $\text{CLE}_{4}$ and its