Random Walk Diffusion Model with a Restricted Solid-on-Solid Condition in Higher Dimensions

Abstract

We study a random walk diffusion model under the restricted solid-on-solid (RSOS) condition in d = 3 + 1 and d = 4 + 1 dimensions. A dropped particle is allowed to take a random walk until satisfying the RSOS condition. The surface width W increases as tβ in early time with β = 0.081 ± 0.006 and becomes saturated at Lα with α = 0.29 ± 0.03 in d = 3 + 1. In d = 4 + 1, W (t) grows as 2b ln t at the beginning and becomes saturated at 2a ln L. The dynamic exponent z ≈ 3.79 is obtained from the relation $$z = {a \over b}$$ . This value is slightly less than, but consistent with, the value obtained from the related Langevin equation. Our results imply that the upper critical dimension of the model is d =4+1.