Exploration trees and conformal loop ensembles

Sheffield⋆, Scott

doi:10.1215/00127094-2009-007

Cited by 194 publications

(384 citation statements)

References 17 publications

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“…When δ < 1, however, (2.2) cannot hold beyond times at which X t = 0 without a so-called principal value correction, because the integral in (2.2) is almost surely infinite beyond such times (see [30,Section 3.1] for additional discussion of this point). Bessel processes can be defined for all time whenever δ > 0 but they are not semi-martingales when δ ∈ (0, 1).…”

Section: Definition Of Sle κ (ρ)mentioning

confidence: 99%

“…is absolutely continuous with 5 We note that [30,Proposition 3.3] states that if X is a continuous process which is adapted to the filtration of a Brownian motion B, solves the Bessel SDE of a fixed dimension δ ∈ (0, 2) when it is not hitting 0, and is instantaneously reflecting at 0, then X has the law of a Bessel process of dimension δ. This result holds under more generality, namely one may assume that for every stopping time τ for X such that X τ = 0 almost surely with σ = inf{t ≥ τ : X t = 0} we have that X | [τ,σ ] is adapted to the filtration…”

Section: Proof Of Conditionmentioning

confidence: 99%

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Imaginary geometry I: interacting SLEs

Miller

Sheffield⋆

2016

Probab. Theory Relat. Fields

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Fix constants χ > 0 and θ ∈ [0, 2π), and let h be an instance of the Gaussian free field on a planar domain. We study flow lines of the vector field e i(h/χ +θ) starting at a fixed boundary point of the domain. Letting θ vary, one obtains a family of curves that look locally like SLE κ processes with κ ∈ (0, 4) (where2 ), which we interpret as the rays of a random geometry with purely imaginary curvature. We extend the fundamental existence and uniqueness results about these paths to the case that the paths intersect the boundary. We also show that flow lines of different angles cross each other at most once but (in contrast to what happens when h is smooth) may bounce off of each other after crossing. Flow lines of the same angle started at different points merge into each other upon intersecting, forming a tree structure. We construct so-called counterflow lines (SLE 16/κ ) within the same geometry using ordered "light cones" of points accessible by angle-restricted trajectories and develop a robust theory of flow and counterflow line interaction. The theory leads to new results about SLE. For example, we prove that SLE κ (ρ) processes are almost surely continuous random curves, even when they intersect the boundary, and establish Duplantier duality for general SLE 16/κ (ρ) processes. Mathematics Subject Classification 60J67B Jason Miller

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Section: Definition Of Sle κ (ρ)mentioning

confidence: 99%

Section: Proof Of Conditionmentioning

confidence: 99%

Imaginary geometry I: interacting SLEs

Miller

Sheffield⋆

2016

Probab. Theory Relat. Fields

Self Cite

208

483

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“…Uniqueness in law then determines the distribution of conditionally onˆ. □ One can use the previous lemma to reconstruct a chordal SLE , ≥ 8 from its dual branches as follows (see also [35] for related considerations).…”

Section: It Follows That (mentioning

confidence: 99%

“…We note that in this context it is rather natural to consider not simply chordal SLE , but a fuller version, such as branching SLE ( [4,35]). …”

Section: It Follows That (mentioning

confidence: 99%

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SLE and the free field: Partition functions and couplings

Dubédat

2009

J. Amer. Math. Soc.

171

276

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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 properties in law. In the present article, some relations between the two objects are studied. We establish identities of partition functions between different versions of SLE and the free field with appropriate boundary conditions; this involves ζ \zeta -regularization and the Polyakov-Alvarez conformal anomaly formula. We proceed with a construction of couplings of SLE with the free field, showing that, in a precise sense, chordal SLE is the solution of a stochastic “differential” equation driven by the free field. Existence, uniqueness in law, and pathwise uniqueness for these SDEs are proved for general κ > 0 \kappa >0 . This identifies SLE curves as local observables of the free field.

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Exponential Decay for the Near‐Critical Scaling Limit of the Planar Ising Model

Camia

Jian

Newman

2020

Comm Pure Appl Math

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We consider the Ising model at its critical temperature with external magnetic field ha 15=8 on the square lattice with lattice spacing a. We show that the truncated two-point function in this model decays exponentially with a rate independent of a as a # 0. As a consequence, we show exponential decay in the near-critical scaling limit Euclidean magnetization field. For the lattice model with a D 1, the mass (inverse correlation length) is of order h 8=15 as h # 0;for the Euclidean field, it equals exactly C h 8=15 for some C . Although there has been much progress in the study of critical scaling limits, results on nearcritical models are far fewer due to the lack of conformal invariance away from the critical point. Our arguments combine lattice and continuum FK representations, including coupled conformal loop and measure ensembles, showing that such ensembles can be useful even in the study of near-critical scaling limits. Thus we provide the first substantial application of measure ensembles.

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Exploration trees and conformal loop ensembles

Cited by 194 publications

References 17 publications

Imaginary geometry I: interacting SLEs

Imaginary geometry I: interacting SLEs

SLE and the free field: Partition functions and couplings

Exponential Decay for the Near‐Critical Scaling Limit of the Planar Ising Model

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