An Almgren monotonicity formula for discrete harmonic functions

Abstract

The celebrated Almgren monotonicity formula for harmonic functions $u:{\mathbb{R}}^n\to \mathbb{R}$ says that its $L^2-$energy concentrated on a sphere of radius $r$, when measured in a suitable sense, is non-decreasing: if $u$ oscillates at a certain scale, it has even larger oscillations at a larger scale. We prove a discrete analogue of the Almgren monotonicity formula for harmonic functions on infinite combinatorial graphs $G=(V,E)$. Some applications are discussed.