Analytical Results for the Statistical Distribution Related to Memoryless Deterministic Tourist Walk: Dimensionality Effect and Mean Field Models

摘要

Consider a medium characterized by N points whose coordinates are randomlygenerated by a uniform distribution along the edges of a unitary d-dimensionalhypercube. A walker leaves from each point of this disordered medium and movesaccording to the deterministic rule to go to the nearest point which has notbeen visited in the preceding \mu steps (deterministic tourist walk). Eachtrajectory generated by this dynamics has an initial non-periodic part of tsteps (transient) and a final periodic part of p steps (attractor). Theneighborhood rank probabilities are parameterized by the normalized incompletebeta function I_d = I_{1/4}[1/2,(d+1)/2]. The joint distributionS_{\mu,d}^{(N)}(t,p) is relevant, and the marginal distributions previouslystudied are particular cases. We show that, for the memory-less deterministictourist walk in the euclidean space, this distribution is:S_{1,d}^{(\infty)}(t,p) = [\Gamma(1+I_d^{-1})(t+I_d^{-1})/\Gamma(t+p+I_d^{-1})] \delta_{p,2}, where t=0,1,2,...,\infty,\Gamma(z) is the gamma function and \delta_{i,j} is the Kronecker's delta. Themean field models are random link model, which corresponds to d \to \infty, andrandom map model which, even for \mu = 0, presents non-trivial cycledistribution [S_{0,rm}^{(N)}(p) \propto p^{-1}]: S_{0,rm}^{(N)}(t,p) =\Gamma(N)/\{\Gamma[N+1-(t+p)]N^{t+p}\}. The fundamental quantities are thenumber of explored points n_e=t+p and I_d. Although the obtained distributionsare simple, they do not follow straightforwardly and they have been validatedby numerical experiments.