FRACTAL MEASURES OF 1ST PASSAGE TIME OF A SIMPLE RANDOM-WALK

We consider random walks, starting at the site i = 1, on a one-dimensi onal lattice segment with an absorbing boundary at i = 0 and a reflect ing boundary at i = L. We find that the typical value of first passage time (FPT) is independent of system size L, while the mean value dive rges linearly with L. The qth moment of the FPT diverges with system s ize as L2q-1, for q > 1/2. For a finite but large L, the FPT distribut ion has an 1/t tail cut off by an exponential of the form exp(-t/L2). However, if L is set equal to infinity, the distribution has an algebr aic tail given by t-3/2. We find that the generalised dimensions D(q) have a nontrivial dependence on q. This shows that the FPT distributio n is a multifractal. We also calculate the singularity spectrum f(alph a).