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doi: 10.6062/jcis.2008.01.01.0003(Free PDF)

Juliana M. Berbert, Rodrigo S. Gonzalez and Alexandre S. Martinez

Consider N points randomly and uniformly distributed in a d-dimensional hypercube. A walker explores this disordered medium going to the nearest site, which has not been visited in the last µ (memory) steps. The walker trajectory is composed of a transient part and a periodic part (cycles). In this case, travelers can or cannot explore all available space, given rise to a crossover at critical memory, for one-dimensional systems µ1 = ln2 N, between localized and extended regimes. The deterministic rule can be softened to consider more realistic situations with the inclusion of a stochastic parameter T (temperature). In this case, the walker movement is defined by a probability density function (PDF) that is parameterized by T and a cost function, which increases as the distance among sites increases. As the temperature increases, the walker can escape from cycles and extend the exploration. Here we review the analytical results obtained for a system with arbitrary dimensionality d for µ = 0 and one-dimensional systems with µ = 1. Also we suggest an extension of this system to study the influence of the temperature on the critical memory.

Stochastic walk, deterministic walk, non-Markovian processes, glass transition.

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