Conformal invariance of planar loop-erased random walks and uniform spanning trees

By Gregory F. Lawler, Oded Schramm, Wendelin Werner. 2001

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This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain \(D\mathop \subset \limits_ \ne \mathbb{C} \) is equal to the radial SLE2 path. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invarint. Assuming that ∂D is a C 1-simple closed curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano curve, where the tree is wired along a proper are A ⊂ ∂ D, is the chordal SLE8 path in \(\overline D \) joining the endpoints of A. A by-product of this result is that SLE8 is almost surely generated by a continuous path. The result and proofs are not restricted to particular choice of Iattice.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/266c3c1a25dc473a32eaa09a8d7b99f83091bf82

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Citations in the corpus

Conformal weldings of random surfaces: SLE and the quantum gravity zipper (2010, result): “= 8. One could also have inferred the existence of such a relationship from the fact | due to Lawler, Schramm, and Werner | that SLE8 is a continuum scaling limit of uniform spanning tree boundaries [LSW04], and the fact | due to Kenyon | that the winding number \height functions" of uniform sp”
The scaling limits of planar LERW in finitely connected domains. (2006, methodology): “The method given in [10] is used to get a coupling of the driving process for the discrete LERW and that for the continuous LERW such that the two driving processes are uniformly close to each other in probability.”
The Brownian loop soup (2003, methodology): “One can use the convergence of loop-erased random walk to the SLE2 curve [ 7 ].”
Convergence of Ising interfaces to Schramm's SLE curves (2013, methodology): “We identify the scaling limits of γ δ along subsequences following the generalization of the approach from [14], outlined in [17].”
Euler Integrals for Commuting SLEs (2005, methodology): “We then proceed to show how to use the Poissonian structure of the loop-soup to define a natural notion of non-intersection for n chordal SLEκ’s, where 0 < κ ≤ 8/3.”
Lectures on two-dimensional critical percolation (2007, methodology): “nov’s self-contained original paper [40]. • For the convergenceto SLE(6), we follow the strategy outlined by Smirnov in [41] (see also our (we=Lawler-Schramm-Werner) paper on loop-erased random walks [26]). • We will follow some SLE(6) computations from [24, 25]. • For how to deduce the critical exp”
Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees (2013, background): “Suppose that D ⊆ C is (a) SLE8 (b) SLE16 Time Figure 1.15: Simulations of SLEκ′ processes in [−1, 1]2 from i to −i for the indicated values of κ′.”
The dimension of the SLE curves (2002, methodology): “of the dimension of the Brownian frontier, this time through SLE 8/3. • κ=8: Here, SLE 8 is known to be the scaling limit of the uniform Peano curve and SLE 2 that of the loop-erased random walk (cf. [12]). Since these two discrete objects are closely related through Wilson’s algorithm, this shows t”
SLE for theoretical physicists (2005, background): “2; unproven, but see [22] for many consequences; • j = 3: cluster boundaries in the Ising model, however, as yet unproven; • j = 4: BCSOS model of roughening transition (equivalent to the 4-state Potts model and the double dimer model), as yet unproven; also certain level lines of a gaussian random ”
Contour lines of the two-dimensional discrete Gaussian free field (2006, result): “γ(t) does exist and is a continuous path. See [RS05] (κ 6= 8) and [ LSW04 ] (κ = 8).”
Universality in the 2D Ising model and conformal invariance of fermionic observables (2009, methodology): “Lawler, Schramm and Werner used in [31] simpler observables to establish conformal invariance of the scaling limit of the UST interfaces and the LERW curves, and to identify them with Schramm’s SLE curves.”
Loewner chains in the unit disk (2009, methodology): “In fact, they turn out to be very useful tools for the study of conformally invariant scaling limits of some classical statistical 2-dimensional lattice models, see [20 – 24].”