Loewner driving functions for off-critical percolation clusters

Abstract

We numerically study the Loewner driving function U_t of a site percolation cluster boundary on the triangular lattice for p<p_c. It is found that U_t shows a drifted random walk with a finite crossover time. Within this crossover time, the averaged driving function < U_t> shows a scaling behavior -(p_c-p) t^{(ν+1)/2ν} with a superdiffusive fluctuation whereas, beyond the crossover time, the driving function U_t undergoes a normal diffusion with Hurst exponent 1/2 but with the drift velocity proportional to (p_c-p)^ν, where ν= 4/3 is the critical exponent for two-dimensional percolation correlation length. The crossover time diverges as (p_c-p)^{-2ν} as p\to p_c.

4 pages, 7 figures

References in corpus (7)

• 2D growth processes: SLE and Loewner chains
• A Guide to Stochastic Loewner Evolution and its Applications
• Stochastic geometry of critical curves, Schramm-Loewner evolutions, and conformal field theory
• Are Domain Walls in Spin Glasses Described by Stochastic Loewner Evolutions?
• Conformal Curves on $W{O}_3$ Surface
• Nodal Domain Statistics for Quantum Maps, Percolation and SLE
• Conformal Invariance of Iso-height Lines in two-dimensional KPZ Surface