# 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} }

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…

## 12 Citations

### Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

- MathematicsProbability 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)…

### Non-simple SLE curves are not determined by their range

- MathematicsJournal 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…

### CLE PERCOLATIONS

- MathematicsForum 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…

### Two-Curve Green’s Function for 2-SLE: The Interior Case

- MathematicsCommunications 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…

### Planar random-cluster model: scaling relations

- PhysicsForum 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…

### Hypergeometric SLE: Conformal Markov Characterization and Applications

- MathematicsCommunications 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…

### Connection probabilities in the double-dimer model -- the case of two connectivity patterns

- 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…

### Hypergeometric SLE: Conformal Markov Characterization and Applications

- Materials ScienceCommunications 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…

### Conformal Markov Characterization of SLE pairs with $\kappa\in (0,4]$

- Mathematics
- 2017

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…

### Two-Curve Green’s Function for 2-SLE: The Interior Case

- Materials ScienceCommunications 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

### Non-simple SLE curves are not determined by their range

- MathematicsJournal 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…

### Conformal Radii for Conformal Loop Ensembles

- 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…

### CLE PERCOLATIONS

- MathematicsForum 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…

### Liouville quantum gravity as a mating of trees

- 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…

### SLE and the free field: partition functions and couplings

- Mathematics
- 2007

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…

### Intersections of SLE Paths: the double and cut point dimension of SLE

- 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…

### Imaginary geometry III: reversibility of SLE_\kappa\ for \kappa \in (4,8)

- 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…

### Conformal Invariance of Boundary Touching Loops of FK Ising Model

- MathematicsCommunications 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.

### Convergence of the Critical Planar Ising Interfaces to Hypergeometric SLE

- Mathematics
- 2016

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…

### ON BOUNDED-TYPE THIN LOCAL SETS OF THE TWO-DIMENSIONAL GAUSSIAN FREE FIELD

- MathematicsJournal 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…