Discrete harmonic functions on infinite penny graphs

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 Poincaré 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.

Harmonic functions of polynomial growth on infinite penny graphs

“…Following these arguments, Delmotte [11] proved the finite-dimensional property for harmonic functions of polynomial growth on graphs under the assumptions of the volume doubling property and the Poincaré inequality. The following result was proved in [17].…”

“…Theorem 2.4 [17,Theorem 1.2]. Let 𝐺 = (𝑉, 𝐸, 𝐹) be an infinite penny graph with bounded facial degree.…”

“…For any graph (𝑉, 𝐸), we denote by 𝑑 𝑉 (𝑑 in short) the combinatorial distance on the graph. The second author [17] proved that any infinite penny graph with bounded facial degree is quasi-isometric to ℤ 2 , see Definition 2.1, and hence the volume doubling property and the Poincaré inequality hold, see Definition 2.3.…”

“…For a penny graph 𝐺 with bounded facial degree, it was conjectured in [17,Conjecture 3.11] that dim  𝑘 (𝐺) ⩽ 𝐶𝑘, ∀𝑘 ⩾ 1.…”