Scaling and Survival Properties of Random Walks with Absorbing and Moving Walls

We study one-dimensional random walks between an absorbing boundary at the origin and a movable wall on the other end. The wall moves outward only, and only when the walker kicks it. The hopping of the walker has no directional bias, except at the location of the kicked wall. We propose a scaling ansatz for the probability distribution P (x, t) for a walker to be at x at time t as P (x, t) = t−δ(q)−1/2f(x/〈x〉), where δ(q) is a function of the bias hopping probability q on the wall and 〈x 〉 is the mean position of the walker. The scaling ansatz is numerically confirmed. From the numerical finding of 〈x 〉 = Ct1/2 and the confirmed scaling relation, the survival probability of the walker can be shown to decay as t−δ(q) with a continuously varying critical exponent δ(q) = 1/2q − 1/2. By setting up the relation between 〈x 〉 with the absorbing boundary and that without the absorbing boundary, we accurately estimate C as C = pi/2 +