MULTIFRACTAL-LIKE FEATURES OF THE PATHS PROBABILITY-DISTRIBUTION IN THE 2-DIMENSIONAL DIRECTED RANDOM-WALK

We consider the two-dimensional directed random walk with probabilitie s p(x) and p(y) assigned to steps in x- and y-direction, respectively, and analyse the probability distribution for the possible 2(t) paths the walker may follow when it performs a t-step walk. This distributio n is multifractal, since its qth moment has the typical power law beha viour of multifractal distributions unless p(x)=p(y)=1/2. If the value s of p(x) and p(y), are allowed to fluctuate around their average valu es [p(x)] and [p(y)] throughout the lattice, the qth moment averaged o ver the possible realizations of the lattice exhibits the multifractal power law behaviour even when [p(x)]=[p(y)]=1/2. The existence of neg ative fractal dimensions is also analysed.