No abstract

Random walks are stochastic processes formed by successive summation of independent, identically distributed random variables and are one of the most studied topics in probability theory. This contemporary introduction evolved from courses taught at Cornell University and the University of Chicago by the first author, who is one of the most highly regarded researchers in the field of stochastic processes. This text meets the need for a modern reference to the detailed properties of an important class of random walks on the integer lattice. It is suitable for probabilists, mathematicians working in related fields, and for researchers in other disciplines who use random walks in modeling.

We characterize and describe all random subsets K of a given simply connected planar domain (the upper half-plane H, say) which satisfy the "conformal restriction" property, i.e., K connects two fixed boundary points (0 and ∞, say) and the law of K conditioned to remain in a simply connected open subset H of H is identical to that of Φ(K), where Φ is a conformal map from H onto H with Φ(0) = 0 and Φ(∞) = ∞. The construction of this family relies on the stochastic Loewner evolution processes with parameter κ ≤ 8/3 and on their distortion under conformal maps. We show in particular that SLE 8/3 is the only random simple curve satisfying conformal restriction and relate it to the outer boundaries of planar Brownian motion and SLE 6 .

This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain D ~ tC 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 invariant. Assuming that aD is a C 1 -simp1e 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 arc A c aD, is the chordal SLEg path in I5 joining the endpoints of A. A by-product of this result is that SLEg is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.

Intersections of random walks I Gregory F. Lawler. p.cm. --(Probability and its applications) Includes bibliographical references and index.1. Random walks (Mathematics) QA274.73.L38 1991 519.2 --dc20Printed on acid-free paper.