Analytical results for the statistical distribution related to a memoryless deterministic walk: Dimensionality effect and mean-field models

Abstract

Consider a medium characterized by N points whose coordinates are randomly generated by a uniform distribution along the edges of a unitary d-dimensional hypercube. A walker leaves from each point of this disordered medium and moves according to the deterministic rule to go to the nearest point which has not been visited in the preceding mu steps (deterministic tourist walk). Each trajectory generated by this dynamics has an initial nonperiodic part of t steps (transient) and a final periodic part of p steps (attractor). The neighborhood rank probabilities are parametrized by the normalized incomplete beta function Id= I1/4 [1/2, (d+1) /2] . The joint distribution S(N) (mu,d) (t,p) is relevant, and the marginal distributions previously studied are particular cases. We show that, for the memory-less deterministic tourist walk in the euclidean space, this distribution is Sinfinity(1,d) (t,p) = [Gamma (1+ I(-1)(d)) (t+ I(-1)(d) ) /Gamma(t+p+ I(-1)(d)) ] delta(p,2), where t=0, 1,2, ... infinity, Gamma(z) is the gamma function and delta(i,j) is the Kronecker delta. The mean-field models are the random link models, which correspond to d-->infinity, and the random map model which, even for mu=0 , presents nontrivial cycle distribution [ S(N)(0,rm) (p) proportional to p(-1) ] : S(N)(0,rm) (t,p) =Gamma(N)/ {Gamma[N+1- (t+p) ] N( t+p)}. The fundamental quantities are the number of explored points n(e)=t+p and Id. Although the obtained distributions are simple, they do not follow straightforwardly and they have been validated by numerical experiments.