Discrete harmonic functions on infinite penny graphs

TL;DR

This paper investigates discrete harmonic functions on infinite penny graphs, establishing key geometric and analytic properties such as volume doubling, Poincaré inequality, Harnack inequality, and finite dimensionality of polynomial growth solutions.

Contribution

It proves that infinite penny graphs with bounded facial degree satisfy fundamental inequalities and properties for harmonic functions, extending understanding of their analytic structure.

Findings

Volume doubling property holds for such graphs

Harnack inequality for positive harmonic functions is established

Finite dimensionality of polynomial growth harmonic functions is proven

Abstract

In this paper, we study discrete harmonic functions on infinite penny graphs. For an infinite penny graph with bounded facial degree, we prove that the volume doubling property and the Poincar\'e inequality hold, which yields the Harnack inequality for positive harmonic functions. Moreover, we prove that the space of polynomial growth harmonic functions, or ancient solutions of the heat equation, with bounded growth rate has finite dimensional property.