2007/02/01 by Cesar Augusto Sangaletti Tercariol, César Augusto Sangaletti Terçariol, Rodrigo Silva Gonzalez +6
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Diffusion and Search Dynamics #Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Physical sciences #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.48550/arxiv.cond-mat/0702030
7 pages and 5 figures
arxiv created 2007/02/01 · openalex publication_date 2007/02/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider N points randomly distributed along a line segment of unitary length. A walker explores this disordered medium moving according to a partially self-avoiding deterministic walk. The walker, with memory μ, leaves from the leftmost point and moves, at each discrete time step, to the nearest point which has not been visited in the preceding μ steps. Using open boundary conditions, we have calculated analytically the probability PN(μ) = (1 - 2-μ)N - μ- 1 that all N points are visited, with N ≫ μ≫ 1. This approximated expression for PN(μ) is reasonable even for small N and μ values, as validated by Monte Carlo simulations. We show the existence of a critical memory μ1 = ln N/ln 2. For μ< μ1 - e/(2ln2), the walker gets trapped in cycles and does not fully explore the system. For μ> μ1 + e/(2ln2) the walker explores the whole system. Since the intermediate region increases as ln N and its width is constant, a sharp transition is obtained for one-dimensional large systems. This means that the walker needs not to have full memory of its trajectory to explore the whole system. Instead, it suffices to have memory of order log2 N.