Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2005-09-07
J. Stat. Mech. (2006) P01001
Physics
Condensed Matter
Statistical Mechanics
Published version, minor typos corrected, added references
Scientific paper
10.1088/1742-5468/2006/01/P01001
Standard 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 Levy process. The situation is defined by the usual SLE parameter, $\kappa$, as well as $\alpha$ which defines the shape of the stable Levy distribution. The resulting behavior is characterized by two descriptors: $p$, the probability that the trace self-intersects, and $\tilde{p}$, 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 $\kappa$ and $\alpha$. It is reasonable to call such changes ``phase transitions''. These transitions occur as $\kappa$ passes through four (a well-known result) and as $\alpha$ passes through one (a new result). Numerical simulations are then used to explore the associated touching and near-touching events.
Gruzberg Ilya A.
Kadanoff Leo P.
Oikonomou P.
Rushkin Ilia
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