On Recurrence and Transience of Fractional RandomWalks in Lattices

Abstract

The study of random walks on networks has become a rapidly growing research field, last but not least driven by the increasing interest in the dynamics of online networks. In the development of fast(er) random motion based search strategies a key issue are first passage quantities: How long does it take a walker starting from a site p 0 to reach ‘by chance’ a site p for the first time? Further important are recurrence and transience features of a random walk: A random walker starting at p 0 will he ever reach site p (ever return to p 0)? How often a site is visited? Here we investigate Markovian random walks generated by fractional (Laplacian) generator matrices L\( \frac{\alpha }{2} \) 2 (0 < \( \alpha \) ≤2) where L stands for ‘simple’ Laplacian matrices. This walk we refer to as ‘Fractional Random Walk’ (FRW). In contrast to classical Polya type walks where only local steps to next neighbor sites are possible, the FRW allows nonlocal long-range moves where a remarkably rich dynamics and new features arise. We analyze recurrence and transience features of the FRW on infinite d-dimensional simple cubic lattices. We deduce by means of lattice Green’s function (probability generating functions) the mean residence times (MRT) of the walker at preselected sites. For the infinite 1D lattice (infinite ring) we obtain for the transient regime (0 < \( \alpha \) < 1) closed form expressions for these characteristics. The lattice Green’s function on infinite lattices existing in the transient regime fulfills Riesz potential asymptotics being a landmark of anomalous diffusion, i.e. random motion (Levy flights) where the step lengths are drawn from a Levy \( \alpha \)-stable distribution.