I Rushkin et al J. Stat. Mech. (2006) P01001 doi:10.1088/1742-5468/2006/01/P01001
Stochastic Loewner evolution driven by Lévy processes
I Rushkin, P Oikonomou, L P Kadanoff and I A Gruzberg1
Show affiliationsStandard stochastic Loewner evolution (SLE) is driven by a continuous Brownian motion, which then produces a continuous fractal trace. If jumps are added to the driving function, the trace branches. We consider a generalized SLE driven by a superposition of a Brownian motion and a stable Lévy process. The situation is defined by the usual SLE parameter, κ, as well as α, which defines the shape of the stable Lévy distribution. The resulting behaviour is characterized by two descriptors: p, the probability that the trace self-intersects, and 
, the probability that it will approach arbitrarily close to doing so. Using Dynkin's formula, these descriptors are shown to change qualitatively and singularly at critical values of κ and α. It is reasonable to call such changes 'phase transitions'. These transitions occur as κ passes through four (a well-known result) and as α passes through one (a new result). Numerical simulations are then used to explore the associated touching and near-touching events.
- Keywords
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E-print Number: cond-mat/0509187
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- PACS
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05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.)
02.60.Cb Numerical simulation; solution of equations
- MSC
- Subjects
- Dates
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Issue 01 (January 2006)
Received 14 September 2005, accepted for publication 1 December 2005
Published 3 January 2006
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Stochastic Loewner evolution driven by Lévy processes
I Rushkin et al J. Stat. Mech. (2006) P01001
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