On the Rate of Convergence of Loop-Erased Random Walk to SLE2

Abstract

We derive a rate of convergence of the Loewner driving function for planar loop-erased random walk to Brownian motion with speed 2 on the unit circle, the Loewner driving function for radial SLE 2 . The proof uses a new estimate of the difference between the discrete and continuous Green's functions that is an improvement over existing results for the class of domains we consider. Using the rate for the driving process convergence along with additional information about SLE 2 , we also obtain a rate of convergence for the paths with respect to the Hausdorff distance.

Regularity of Loewner Curves

“…As said in the introduction, these types of convergence do not directly involve the curves. In [4,Section 7], it is shown that if two driving functions generating simple curves are close in the sup-norm and if one function has the condition (3.4) then the two generated curves are close in Hausdorff distance. One really wants two curves to be close in the sup-norm.…”

“…However these types of convergence relate to Loewner chains rather than curves, see [14,Chapter 4] and respectively [2] for details. For κ ≤ 4, when one views curves as compact sets, the sequence γ n converges almost surely to SLE κ in Hausdorff metric [4,Section 7]. A general principle is to set up a theorem for the deterministic Loewner equation and then translate the result into the SLE context.…”

Transition probabilities for infinite two-sided loop-erased random walks

“…We let T + be the hitting time of [0, ∞) by B t . Using the Beurling estimate (see [12] for the discrete version, [7] for a discussion of the continuous case) in an argument very similar to that in Proposition 3.1 in [1], one can show that…”

“…w∈Cm h η m (z, w)Using the previous lemma, we see thatw∈Cm h η m (z, w) H q An (w, −n) = O(m − 3 2 −u ) + v η (z) w∈Cm µ m (w) H q An (w, −n) m 1Note that symmetry and the argument of Lemma 3.8 imply thatw∈Cm µ m (w)H q An (w, −n) = w∈Cm P −1 (S(σ m ∧ τ + ) = w) P −1 (σ m < τ + ) H A − n (w, −n) = w∈Cm,d(w,m)≥m 3/−1 (S(σ m ∧ τ + ) = w) 4P −1 (σ m < τ + ) ·G A − n (w, −(n − 1))(1 + O(n −1/2 )). The argument of Lemma 3.8 and Theorem 8.1 in[1] imply thatµ m (w) H q An (w, −n) = 1 2π h (−1, w)g A − n (w, −(n − 1))(1 + O(n −1/2+ǫ )) dw,whereh is the Brownian Poisson kernel in the slit disk conditional on not leaving at the slit and the integral is over {w : |w| = m, d(w, m) ≥ R 3/4 }. This last expression can be shown to equal c ′ n [1 + O(n −u )] for some c ′ .…”

The Difference Between a Discrete and Continuous Harmonic Measure

“…for all functions g on ∂D which are C 2+α with respect to arc length along the boundary for some α > 0. ω h is defined in (2), and ω is defined in (1).…”