Scaling limit of the loop-erased random walk Green’s function

Abstract

We consider loop-erased random walk (LERW) running between two boundary points of a square grid approximation of a planar simply connected domain. The LERW Green's function is the probability that the LERW passes through a given edge in the domain. We prove that this probability, multiplied by the inverse mesh size to the power 3/4, converges in the lattice size scaling limit to (a constant times) an explicit conformally covariant quantity which coincides with the SLE 2 Green's function.The proof does not use SLE techniques and is based on a combinatorial identity which reduces the problem to obtaining sharp asymptotics for two quantities: the loop measure of random walk loops of odd winding number about a branch point near the marked edge and a "spinor" observable for random walk started from one of the vertices of the marked edge.

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

“…We will compute the transition probability by writing the expression for the strip A n and then taking the limit as n → ∞. This argument will only use the expressions derived in [2] and random walk estimates. We will first restate the main result for d = 2, and then we will define the quantities in the statement.…”

“…, R], where U R is as in (8). This is because the slit square can be split into a finite number of rectangles and the (discrete) Poisson kernel for a rectangle can be given explicitly; this idea is used in Section 5 of [2] and we state some of the main results here.…”

“…We are left with estimate the quantity for Brownian motion which can be done using conformal invariance. The Poisson kernel in the slit disk is (see for instance the proof of Lemma 5.1 in [2]) h D\[0,1) (ǫe iµ , e iθ ) = 1 2π ǫ 1/2 sin(µ/2) sin(θ/2) + O(ǫ).…”

“…This paper was motivated by a question from Kenyon and Wilson when they were preparing a paper on spanning trees with a two-sided backbone [14]. At the time, we thought that our results [2] immediately gave a determinantal formula (a version of Fomin's identity) for the two-sided walk. It turned out that taking the limit, and, in particular, showing independence of boundary conditions required more work.…”

“…In this case, the marked edge is on the boundary of the slit domain and there is no need to use signed weights. The reader may also compare the formula of Theorem 1.2 with Theorem 3.1 of [2] which gives the probability that a chordal LERW visits a particular interior edge in terms of a determinant involving signed Poisson kernels. In fact, that formula is used in the proof of Theorem 1.2.…”