Multi-headed lattices and Green functions

TL;DR

This paper investigates Green functions on multi-headed lattices, analyzing differential equations and Pólya numbers, and completes the classification for certain low-dimensional cases using creative telescoping.

Contribution

It introduces a novel application of creative telescoping to analyze Green functions on multi-headed lattices, filling gaps in low-dimensional cases.

Findings

Complete analysis of three missing cases in dimensions up to five.

Derived differential equations for Green functions on these lattices.

Calculated Pólya numbers for the studied lattice geometries.

Abstract

Lattice geometries and random walks on them are of great interest for their applications in different fields such as physics, chemistry, and computer science. In this work, we focus on multi-headed lattices and study properties of the Green functions for these lattices such as the associated differential equations and the P\'olya numbers. In particular, we complete the analysis of three missing cases in dimensions no larger than five. Our results are built upon an automatic machinery of creative telescoping.