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

TL;DR

This paper derives analytical distributions for the memoryless deterministic tourist walk in high-dimensional spaces, revealing how dimensionality influences transient and cycle lengths, with validation through numerical experiments.

Contribution

It provides exact analytical formulas for the joint distribution of transient and cycle lengths in the tourist walk model, including mean field approximations and their validation.

Findings

Distribution simplifies to a gamma function expression for infinite dimensions.

Mean field models include the random link and random map models with distinct cycle distributions.

Distributions are validated by numerical experiments despite their complexity.

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 non-periodic part of t steps (transient) and a final periodic part of p steps (attractor). The neighborhood rank probabilities are parameterized by the normalized incomplete beta function I_d = I_{1/4}[1/2,(d+1)/2]. The joint distribution S_{\mu,d}^{(N)}(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: 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. The mean field models are random link model, which corresponds to d \to \infty, and random map model which, even for \mu = 0, presents non-trivial cycle distribution [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 the number of explored points n_e=t+p and I_d. Although the obtained distributions are simple, they do not follow straightforwardly and they have been validated by numerical experiments.