Fractal measures of first passage time of a simple random walk

We consider random walks, starting at the site i = 1, on a one-dimensional lattice segment with an absorbing boundary at i = 0 and a reflecting 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 diverges linearly with L. The qth moment of the FPT diverges with system size as L2q−1, for q >12. For a finite but large L, the FPT distribution 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 algebraic tail given by t-12. We find that the generalised dimensions D(q) have a nontrivial dependence on q. This shows that the FPT distribution is a multifractal. We also calculate the singularity spectrum f(α).