On the field dependence of random walks in the presence of random fields

Numerical simulations and scaling arguments are used to study the field dependence of a random walk in a one-dimensional system with a bias field on each site. The bias is taken randomly with equal probability to be +E or -E. The probability density -P(x, t) is found to scale asymptotically as {Mathematical expression} with A(E)=ln[(1+E)/(1-E)], β=4.25, and α=1.25. The mean square displacement scales as {Mathematical expression}, where F(u)∼ln⁴u asymptotically.