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.

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.

We use elementary methods to derive a sharp local central limit theorem for simple random walk on the square lattice in dimensions one and two that is an improvement of existing results for points that are particularly distant from the starting point. More specifically, we give explicit asymptotic expressions in terms of n and x in dimensions one and two for P (S(n) = x), the probability that simple random walk S is at some point x at time n, that are valid for all |x| ≤ n/ log 2 (n). We also show that the behavior of planar simple random walk differs radically from that of planar standard Brownian motion outside of the disk of radius n 3/4 , where simple random walk ceases to be approximately rotationally symmetric. Indeed, if n 3/4 = o(|S(n)|), S(n) is more likely to be found along the coordinate axes. This loss of rotational symmetry is not surprising, since if |S(n)| = n, there are only four possible locations for S(n). In this paper, we show how the transition from approximate rotational symmetry to complete concentration of S along the coordinate axes occurs.