Fetching the paper…

Reading the bibliography…

2016

Gaussian free field light cones and SLE$_\kappa(\rho)$

Miller, Jason, Sheffield, Scott

Original
PDF
Post on X

Understand

We derive a surprising correspondence between SLE$_{\kappa}(\rho)$ processes and light cones of the Gaussian free field (GFF).

Recall that (one-sided, chordal, origin-seeded) SLE$_\kappa(\rho)$ processes are in some sense the simplest and most natural variants of the Schramm-Loewner evolution.
They were originally defined only for $\rho > -2$, but one can use L\'evy compensation to extend the definition to any $\rho > -2-\tfrac{\kappa}{2}$ and to obtain qualitatively different curves.
The triangle $T = \{(\kappa, \rho): (-2-\tfrac{\kappa}{2})\vee (\tfrac{\kappa}{2}-4) < \rho < -2 \}$ is the primary focus of this paper.

Gaussian free field light cones and SLE$_\kappa(\rho)$ · Around

Built on

Continuous martingales and Brownian motion
D. Revuz and M. Yor · 1999
Earlier work this paper cites.
Scaling limits of loop-erased random walks and uniform spanning trees
O. Schramm · 2000
Earlier work this paper cites.
Conformal restriction: the chordal case
G. Lawler, O. Schramm, and W. Werner · 2003
Earlier work this paper cites.
Basic properties of SLE
S. Rohde and O. Schramm · 2005
Earlier work this paper cites.
Gaussian free fields for mathematicians
S. Sheffield · 2007
Earlier work this paper cites.
Duality of chordal SLE
Original
D. Zhan · 2008
Earlier work this paper cites.

Similar

Duality of Schramm-Loewner evolutions
Original
J. Dubédat · 2009
Cited alongside, same era.
SLE and the free field: partition functions and couplings
Original
J. Dubédat · 2009
Cited alongside, same era.
Exploration trees and conformal loop ensembles
S. Sheffield · 2009
Cited alongside, same era.
Conformal weldings of random surfaces: SLE and the quantum gravity zipper
Original
S. Sheffield · 2010
Cited alongside, same era.
Duality of chordal SLE, II
Original
D. Zhan · 2010
Cited alongside, same era.

Then

Imaginary Geometry I: Interacting SLEs
Original
J. Miller and S. Sheffield · 2012
Later among the works it cites.
Imaginary geometry II: reversibility of SLE ( ρ 1 ; ρ 2 ) κ {}_{\kappa}(\rho_{1};\rho_{2}) for κ ∈ ( 0 , 4 ) \kappa\in(0,4)
Original
J. Miller and S. Sheffield · 2012
Later among the works it cites.
Imaginary geometry III: reversibility of SLE κ
Original
J. Miller and S. Sheffield · 2012
Later among the works it cites.
Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees
Original
J. Miller and S. Sheffield · 2013
Later among the works it cites.
A contour line of the continuum Gaussian free field
Original

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

Reversibility of some chordal SLE ⁡ ( κ , ρ ) {\rm SLE}(\kappa;\rho) traces

D. Zhan · 2010

Cited alongside, same era.

Dimension of the SLE light cone, the SLE fan, and SLE ( ρ ) κ {}_{\kappa}(\rho) for κ ∈ ( 0 , 4 ) \kappa\in(0,4) and ρ ∈ [ κ 2 − 4 , − 2 ) \rho\in[\tfrac{\kappa}{2}-4,-2)

J. Miller

Cited in the paper.

Local sets of the Gaussian free field: slides and audio. www.fields.utoronto.ca/0506/percolationsle/sheffield1, www.fields.utoronto.ca/audio/0506/percolationsle/sheffield2, www.fields.utoronto.ca/audio/0506/percolationsle/sheffield3

S. Sheffield

Cited in the paper.

O. Schramm and S. Sheffield · 2013

Later among the works it cites.

CLE percolations

Original

J. Miller, S. Sheffield, and W. Werner · 2016

Closest in time.