Conformal restriction: The chordal case

By Gregory F. Lawler, Oded Schramm, Wendelin Werner. 2002

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We characterize and describe all random subsets K of a given simply connected planar domain (the upper half-plane Η, say) which satisfy the conformal restriction” property, i.e., K connects two fixed boundary points (0 and ∞, say) and the law of K conditioned to remain in a simply connected open subset H of H is identical to that of Φ( K ), where Φ is a conformal map from H onto H with Φ(0) = 0 and Φ(∞) = ∞. The construction of this family relies on the stochastic Loewner evolution processes with parameter κ ≤ 8/3 and on their distortion under conformal maps. We show in particular that SLE 8/3 is the only random simple curve satisfying conformal restriction and relate it to the outer boundaries of planar Brownian motion and SLE 6 . See link below.

Subfields: Applied Mathematics, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/23d20dc1e07930f0c70fd850faebf60b1764db94

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Citations in the corpus

Reversibility of chordal SLE (2008, background): “Then (Kj(t),0 ≤ t < Tj) are an SLE(κ,κ− 6) process [5] started from xj with force point at xk; Tj is the first time that xk is swallowed by Kj(t); and φj(t, xk) = pj(t), 0≤ t < Tj .”
Duality of chordal SLE (2007, background): “, the convergence of self-avoiding walk to SLE(8/3) [8], and double domino tilling to SLE(4) [11].”
The energy of a deterministic Loewner chain: Reversibility and interpretation via SLE$_{0+}$ (2016, background): “1 Conformal restriction In this subsection we study the variation of the energy of a given curve when the domain varies (this is similar to the conformal restriction-type ideas initiated in [12] in the SLE-framework, similar computations can be also found in the Section 9.”
Imaginary geometry I: interacting SLEs (2012, background): “3], however a proof of this fact is not given in [15].”
Liouville quantum gravity and KPZ (2008, background): “Wi > κ/2 corresponds to the condition that the ρi are in the range for which the path almost surely does not intersect the boundary [LSW03].”
Conformally invariant scaling limits: an overview and a collection of problems (2006, background): “It is not clear if one can make good sense of that.”
The dimension of the SLE curves (2002, background): “This result was known for 8 (because the curve is then space- lling), = 6 (see [2], recall that this corresponds to the scaling limit of critical percolation clusters) and = 8=3 (this follows from the description of the outer frontier of SLE 6 or planar Brownian motion in terms of SLE 8=3 in [8], an”
Excursion decompositions for SLE and Watts' crossing formula (2004, background): “The SLE(κ, ρ) processes were introduced in [11]; see also [25, 7].”
Harmonic explorer and its convergence to SLE4 (2003, background): “A slightly different variant of the continuous Brownian excursion was studied in [8].”
Basic properties of SLE (2001, background): “Motivated in part by Duplantier’s duality conjecture 9.5, in [ LSW03 ] it is conjectured that chordal SLE8/3 is the scaling limit of the self-avoiding walk”
SLE and the free field: partition functions and couplings (2007, methodology): “This is proved in Proposition 3 in [10], based on results in [22].”
Multiple Schramm–Loewner Evolutions and Statistical Mechanics Martingales (2005, neutral): “…. . . . . . . . . . 31 8.2.1 3SLE (pure) configurations . . . . . . . . . . . . . . . 31 8.2.2 Arch probabilities . . . . . . . . . . . . . . . . . . . . 33 8.3 Applications to percolation and Ising model . . . . . . . . . . 35 8.4 nSLEs and beyond . . . . . . . . . . . . . . . . . . . . . . . . 36”
The scaling limits of planar LERW in finitely connected domains. (2006, background): “In [9], SLE8/3 is proved to have the restriction property, and so is conjectured to be the scaling limits of self-avoiding walk.”
Recent advances in percolation theory and its applications (2015, background): “SLEκ has the restriction property only for κ = 8/3 and no other values of κ [263].”