Year: 2009
Type: preprint
Abstract: The Schramm-Loewner evolution (SLE_κ) is a candidate for the scaling limit of random curves arising in two-dimensional critical phenomena. When κ< 8, an instance of SLE_κis a random planar curve with almost sure Hausdorff dimension d = 1 + κ/8 < 2. This curve is conventionally parametrized by its half plane capacity, rather than by any measure of its d-dimensional volume. For κ< 8, we use a Doob-Meyer decomposition to construct the unique (under mild assumptions) Markovian parametrization of SLE_κthat transforms like a d-dimensional volume measure under conformal maps. We prove that this parametrization is non-trivial (i.e., the curve is not entirely traversed in zero time) for κ< 4(7 - \sqrt{33}) = 5.021 ....
Source: arXiv (Cornell University)
Authors: Lawler, Gregory F., Sheffield, Scott
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2102558769