We define multiple chordal SLEs in a simply connected domain by considering a natural configurational measure on paths. We show how to construct these measures so that they are conformally covariant and satisfy certain boundary perturbation and Markov properties, as well as a cascade relation. As an example of our construction, we derive the scaling limit of Fomin's identity in the case of two paths directly; that is, we prove that the probability that an SLE 2 and a Brownian excursion do not intersect can be given in terms of the determinant of the excursion hitting matrix. Finally, we define the λ-SAW, a one-parameter family of measures on self-avoiding walks on Z 2 .
We prove an estimate for the probability that a simple random walk in a simply connected subset A ⊂ Z 2 starting on the boundary exits A at another specified boundary point. The estimates are uniform over all domains of a given inradius. We apply these estimates to prove a conjecture of S. Fomin [4] in 2001 concerning a relationship between crossing probabilities of loop-erased random walk and Brownian motion.
We derive a number of estimates for the probability that a chordal SLE κ path in the upper half plane H intersects a semicircle centred on the real line. We prove that if 0 < κ < 8 andwhere a = 2/κ and C(x; rx) denotes the semicircle centred at x > 0 of radius rx, 0 < r ≤ 1/3, in the upper half plane. As an application of our results, for 0 < κ < 8, we derive an estimate for the diameter of a chordal SLE κ path in H between two real boundary points 0 and x > 0. For 4 < κ < 8, we also estimate the probability that an entire semicircle on the real line is swallowed at once by a chordal SLE κ path in H from 0 to ∞.2000 Mathematics Subject Classification. 82B21, 60K35, 60G99, 60J65
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.