Loewner driving functions for off-critical percolation clusters

TL;DR

This paper investigates the behavior of the Loewner driving function for off-critical percolation clusters, revealing a crossover from superdiffusive to diffusive regimes and characterizing the scaling behavior near criticality.

Contribution

It provides a numerical analysis of the Loewner driving function in off-critical percolation, identifying the crossover dynamics and scaling laws related to the correlation length exponent.

Findings

Driving function exhibits a drifted random walk with a finite crossover time.

Within crossover time, the average driving function scales as -(p_c-p) t^{( u +1)/2 u}.

Beyond crossover, the driving function diffuses normally with a drift velocity proportional to (p_c-p)^ u.

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^{(\nu +1)/2\nu} 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)^\nu, where \nu= 4/3 is the critical exponent for two-dimensional percolation correlation length. The crossover time diverges as (p_c-p)^{-2\nu} as p\to p_c.