Loewner driving functions for off-critical percolation clusters

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→p_c.