| Summary: | This dissertation investigates properties of the Schramm--Loewner evolution (SLE), a family of conformally invariant random planar curves that arise as universal scaling limits of discrete models from probability and statistical physics.
The unifying thread of this work is to consider SLE as a measure on curves rather than simply a stochastic process, so as to carry over some of the intuition and techniques from statistical physics to the continuum. We restrict our study mostly to the parameter values kappa ≤ 4, for which the curves do not touch themselves or the boundary.
First, we give a description of the reversal of radial SLE in terms of the natural base measure for curves that start at an interior point, which is whole-plane SLE. We give an explicit formula for the Radon--Nikodym derivative between these two measures. In doing so we establish a strong estimate (joint with Greg Lawler) on the renormalized Brownian loop measure of the loops that hit two disjoint sets.
Second, we provide uniform estimates, developed jointly with Greg Lawler, on the probability that a radial or two-sided radial SLE curve, which is targeted at a specific interior point, returns far from that point after first coming close to it. These estimates are related to questions about the transience or endpoint continuity of SLE curves and can be considered as quantitative versions of those results.
Third, we prove a global relationship between two-sided radial SLE and chordal SLE. Two-sided radial SLE can be thought of roughly as chordal SLE restricted to curves passing through a marked point (appropriately normalized). The theorem states that the aggregate (integral) of two-sided radial SLE measures over all possible marked points is exactly chordal SLE biased by the total length of the curve in the natural parametrization for SLE. This theorem is an initial result for SLE in the direction of the general rooting and unrooting of measures on curves, which has been well studied for Brownian motion.
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