Random walk weakly attracted to a wall

TL;DR

This paper analyzes a one-dimensional random walk with a bias near a wall, showing its expected position scales as a power law with exponent depending on a parameter, using spectral methods.

Contribution

It provides an explicit spectral analysis of a biased random walk near a wall, deriving the asymptotic behavior of its expectation.

Findings

Expected position scales as n^{1-(elta/2)} for elta in (1,2)

Spectral representation is explicitly derived for this walk

Provides rigorous proof of asymptotic behavior

Abstract

We consider a random walk X_n in Z_+, starting at X_0=x>= 0, with transition probabilities P(X_{n+1}=X_n+1|X_n=y>=1)=1/2-\delta/(4y+2\delta) P(X_{n+1}=X_n+1|X_n=y>=1)=1/2+\delta/(4y+2\delta) and X_{n+1}=1 whenever X_n=0. We prove that the expectation value of X_n behaves like n^{1-(\delta/2)} as n goes to infinity when \delta is in the range (1,2). The proof is based upon the Karlin-McGregor spectral representation, which is made explicit for this random walk.