Multifractal-like features of the paths probability distribution in the two-dimensional directed random walk

We consider the two-dimensional directed random walk with probabilities px and py assigned to steps in x- and y-direction, respectively, and analyse the probability distribution for the possible 2t paths the walker may follow when it performs a t-step walk. This distribution is multifractal, since its qth moment has the typical power law behaviour of multifractal distributions unless . If the values of px and py are allowed to fluctuate around their average values and throughout the lattice, the qth moment averaged over the possible realizations of the lattice exhibits the multifractal power law behaviour even when . The existence of negative fractal dimensions is also analysed.