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

TL;DR

This paper investigates the probability distribution of sums of positions of a restricted random walk, demonstrating that under certain conditions, the distribution closely follows q-Gaussian forms, providing insights into non-extensive statistical mechanics.

Contribution

It introduces exact solutions for the restricted random walk model and shows the conditions under which q-Gaussians accurately describe the distribution, extending the applicability of q-statistics.

Findings

P(y,T) is well approximated by q-Gaussians at T*~L^2

The q-Gaussian fit remains high quality for various transition exponents a

Deviations from q-Gaussian occur when a≠1

Abstract

We present exact results obtained from Master Equations for the probability function P(y,T) of sums $y=\sum_{t=1}^T x_t$ 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 $T^*\propto L^2$. 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, all though weak, but essential, deviation from the q-Gaussian does occur for $a\neq1$. 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).