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 µ 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 functionis 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:is the gamma function and δi,j is the Kronecker's delta. The mean field models are random link model, which corresponds to d → ∞, and random map model which, even for µ = 0, presents non-trivial cycle distribution [SThe fundamental quantities are the number of explored points ne = t + p and I d . Although the obtained distributions are simple, they do not follow straightforwardly and they have been validated by numerical experiments.

Ergodic crossover in partially self-avoiding stochastic walks

“…We have shown [8] that if μ < μ 1 , the walker is trapped in small regions. Nevertheless, if μ μ 1 , the walker crosses the medium, if leaving from one extremity [8].…”

“…We have shown [8] that if μ < μ 1 , the walker is trapped in small regions. Nevertheless, if μ μ 1 , the walker crosses the medium, if leaving from one extremity [8]. As an optimization problem, for small memories, compared to system size, one guarantees the whole landscape exploration [8,17,18].…”

“…For instance, consider a deterministic partially self-avoiding walk [5][6][7][8] in which a walker can visit N sites, randomly distributed on a landscape, following the deterministic rule of going to the nearest site not visited in the last μ steps, including the current site [5]. One can compare this walker with a tourist exploring cities, and the memory μ with the time required for a city already visited to become attractive again to the tourist.…”

“…Nevertheless, if μ μ 1 , the walker crosses the medium, if leaving from one extremity [8]. As an optimization problem, for small memories, compared to system size, one guarantees the whole landscape exploration [8,17,18]. The robustness of this crossover has been addressed with the inclusion of a stochastic factor.…”

Color Texture Analysis and Classification: An Agent Approach Based on Partially Self-avoiding Deterministic Walks

“…After a very short transient time, the walker becomes trapped by a couple of mutually nearest neighbors. The transient time and period joint probability distribution, for N 1, is [19]: [20]:…”