On some problems concerning planar random walks

We consider two questions related to planar simple random walk. Our first result confirms an observation made by Mandelbrot about the number of large holes made by planar simple random walk S. We show that if Ndelta is the number of components of C S[0, 2n] of area greater than n1-delta, then for all delta less than or equal to some delta0 > 0, log2 nd ndNd→ P2p,asn →infinity; In the second part of the thesis, we establish some of the basic estimates needed to extend the main result of [23], where it is shown that the scaling limit of loop-erased random walk from an interior point of a domain to the boundary is the radial Schramm-Lowner Evolution with parameter 2 ( SLE2), to the chordal case. The expected result is that the scaling limit of loop-erased random walk (LERW) excursion in the upper half-plane H is chordal SLE2. The natural time parameter for chordal SLE is the half-plane capacity, heap, as introduced in [17]. We define the discrete half-plane capacity, denoted by dhcap, an analogous quantity for discrete subsets of the discrete upper half-plane H , as well as a natural correspondence between discrete sets A ⊂ H and continuous sets A ⊂ H . We show that for a large class of such sets dhcap(A) is close to hcap(A). We estimate very precisely a discrete Green's function in H A and express it in terms of various parameters of the set A. Applying this to LERW excursion and using the relationship between heap and dhcap should provide information on the driving process of the LERW path, whose scaling limit is expected to be Brownian motion with variance 2. In the event that the scaling limit of LERW excursion is indeed chordal SLE2, this work should give us the necessary background to study another question: how does SLE relate to general Laplacian random walks, a family of random walks of which LERW is a special case?; In both problems we work with coupling methods for random walk and Brownian motion, namely Skorokhod embedding and the so-called KMT approximation. Other tools used are bounds on derivatives of conformal transformations, the Beurling projection theorem, and ideas involving Brownian disconnection exponents.