Monotonicity formulas are among the most powerful tools in the analysis of elliptic PDEs. Almgren's frequency formula — the ratio of a harmonic function's energy on a sphere to its L² norm on that sphere increases with the radius — controls the vanishing order of solutions and drives regularity theory. These formulas are inherently continuous: they use integrals over spheres, derivatives with respect to radius, and the smooth structure of Euclidean space.
Atwood, Smit Vega Garcia, and Wang (arXiv:2603.13132) establish the discrete analogs on infinite d-regular trees. Harmonic functions on trees satisfy a mean-value property (the value at a vertex equals the average over its neighbors), and the tree's geometry provides a natural notion of “sphere” (all vertices at graph distance r from the root) and “ball” (all vertices at distance at most r).
Three monotonicity formulas transfer: a weighted Dirichlet energy (the sum of squared differences across edges at distance r), a Weiss-type formula (combining energy and boundary terms with specific weights), and a generalized Almgren formula (the ratio of energy to Lp norm for p ≥ 1). The proofs require adapting continuous calculus — integration by parts, co-area formula, chain rule — to the discrete setting, where sums replace integrals and differences replace derivatives.
The concrete computations on 2-regular and 3-regular trees reveal the transition: on the 2-regular tree (which is just the integer line), the formulas reduce to elementary identities about sequences. On the 3-regular tree, genuine tree structure appears — the branching creates geometric effects absent from the line. The monotonicity persists but the constants change, reflecting the exponential growth of spheres on the tree versus the polynomial growth in Euclidean space.