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Schramm Loewner Evolution and Liouville Quantum Gravity (1012.4800v1)
Published 21 Dec 2010 in math-ph, cond-mat.stat-mech, math.CV, math.MP, and math.PR
Abstract: We conformally weld (via "quantum zipping") two boundary arcs of a Liouville quantum gravity random surface to generate a random curve called the Schramm-Loewner evolution (SLE). We develop a theory of quantum fractal measures (consistent with the Knizhnik-Polyakov-Zamolochikov relation) and analyze their evolution under welding via SLE martingales. As an application, we construct the natural quantum length and boundary intersection measures on the SLE curve itself.
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