The celebrated Almgren monotonicity formula for harmonic functions $u:\mathbb{ R}^n \rightarrow \mathbb{ R}$ 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.
Categories: math.AP math.PR