A Generalisation of the Sparre Andersen Theorem on Fractal Structures

The Sparre Andersen theorem establishes the asymptotical time
dependence of the first passage time distribution, in the case of a one
dimensional random walk, as t^{−3/2} . We generalize this result to
the case of a random walk on an infinite fractal structure, resulting
in a time dependence of the form t^{ds /2−2} , where ds is the
spectral dimension of the walk. This result is verified numerically and
also leads to an expression relating the prefactors of the time
dependence of the probability of being at the origin and the first
passage time distribution. A more detailed analysis results in the
distance dependent asymptotic form of the first passage time
distribution, r^{dw −df} t^{ds /2−2} , where dw is the walk dimension
and df the fractal dimension. The distance dependence is interestingly
found to match that of a random walk in a confined domain, refining the
understanding of compact exploration.