Restricted random walk model as a new testing ground for the applicability of q-statistics

¹ Department of Physics, Faculty of Science, Ege University - 35100 Izmir, Turkey
² Complexity & Networks Group and Department of Mathematics, Imperial College London, South Kensington Campus - London SW7 2AZ, UK, EU
³ Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil
⁴ Santa Fe Institute - 1399 Hyde Park Road, Santa Fe, NM 87501, USA

^a h.jensen@imperial.ac.uk

Received: 8 July 2011
Accepted: 26 September 2011

Abstract

We present exact results obtained from Master Equations for the probability function P(y, T) of sums of the positions x_t of a discrete random walker restricted to the set of integers between −L and L. We study the asymptotic properties for large values of L and T. For a set of position-dependent transition probabilities the functional form of P(y, T) is with very high precision represented by q-Gaussians when T assumes a certain value T^*∝L². The domain of y values for which the q-Gaussian apply diverges with L. The fit to a q-Gaussian remains of very high quality even when the exponent a of the transition probability g(x)=|x/L|^a+p with 0<p≪1 is different from 1, although weak, but essential, deviation from the q-Gaussian does occur for a≠1. To assess the role of correlations we compare the T dependence of P(y, T) for the restricted random walker case with the equivalent dependence for a sum y of uncorrelated variables x each distributed according to 1/g(x).