Random Walk in Periodic Environment

TL;DR

This paper studies a specific type of random walk in a periodic environment on Z^d, establishing fundamental probabilistic laws and geometric properties of the walk's asymptotic behavior under certain conditions.

Contribution

It demonstrates that the law of large numbers and central limit theorem apply to periodic environment RWRE, and characterizes the angle between the walk's direction and the potential gradient in the reversible case.

Findings

LLN and CLT hold for RWPE under natural conditions.

In the reversible case, the asymptotic direction forms an angle less than pi/2 with the potential gradient.

The angle can approach pi/2, indicating potential asymptotic increase along the trajectory.

Abstract

We consider a special case of random walk in random environment (RWRE) on Z^d where the environment is periodic (RWPE). Under natural conditions, we show that law of large numbers and central limit theorem holds. In the ballistic nearest neighbour reversible case, we prove that the angle between the asymptotic direction of the RWPE and the average negative gradient of the potential function of the reversible environment is less than pi/2, that is, the potential cannot increase asymptotically along the trajectory of the RWPE. But this angle can be close to pi/2.