Random walks on finite lattice tubes

TL;DR

This paper provides exact analytical results for random walks on finite lattice tubes, including absorption probabilities and site visitation expectations, applicable to various lattice types and modeling effects like strain and surface curvature.

Contribution

It derives explicit formulas for absorption probabilities and visitation expectations on finite lattice tubes of arbitrary size and lattice type, incorporating strain effects.

Findings

Explicit formulas for absorption probabilities at tube ends

Expected visitation counts for specific lattice sites

Modeling of strain effects on surface diffusion

Abstract

Exact results are obtained for random walks on finite lattice tubes with a single source and absorbing lattice sites at the ends. Explicit formulae are derived for the absorption probabilities at the ends and for the expectations that a random walk will visit a particular lattice site before being absorbed. Results are obtained for lattice tubes of arbitrary size and each of the regular lattice types; square, triangular and honeycomb. The results include an adjustable parameter to model the effects of strain, such as surface curvature, on the surface diffusion. Results for the triangular lattice tubes and the honeycomb lattice tubes model diffusion of adatoms on single walled zig-zag carbon nano-tubes with open ends.