Green geometry, Martin boundary and random walk asymptotics on groups

TL;DR

This paper introduces the Green-variation functional as a key analytic quantity linking Martin boundary behavior, Green geometry, and random walk escape rates on finitely generated groups, providing new criteria and obstructions for boundary collapse and speed properties.

Contribution

It establishes a novel, checkable analytic quantity that characterizes boundary collapse and Green geometry behavior, along with criteria for its vanishing and implications for random walk speed and harmonic functions.

Findings

Green-variation functional characterizes Martin boundary collapse.

Criteria for b4-vanishing based on heat kernel bounds and elliptic conditions.

Obstructions to strong Liouville property on exponential growth groups.

Abstract

We identify a single computationally checkable analytic quantity interlacing Martin boundary collapse, Green geometry, and linear escape for transient random walks on finitely generated groups: the Green-variation functional \[ \Delta(S;a,b):=\max_{x\in\partial S}\frac{|G(a,x)-G(b,x)|}{G(a,x)}. \] We prove that $\Delta\to0$ along exhaustions characterises the strong Liouville property (under mild, verifiable hypotheses on the ``strong Liouville $\Rightarrow \Delta\to0$'' direction), turning boundary oscillation estimates for Green kernels into potential-theoretic rigidity. We then give two general criteria for $\Delta$-vanishing. The first one derives quantitative bounds on $\Delta$ from coarse heat-kernel envelopes at an intrinsic scale together with a Tauberian comparability, covering Gaussian/sub-Gaussian and stable-like regimes; and the second one is purely elliptic: an ``elliptic H\"older exhaustion'' criterion. Conversely, on groups of exponential growth, $\Delta$ fails to decay along balls already under stretched-exponential on-diagonal upper bounds, yielding a quantitative obstruction to strong Liouville. As consequences, trivial Martin boundary forces linear-scale collapse of Green geometry ($d_G(e,x)=o(|x|)$) and vanishing Green speed (in probability), without any entropy hypothesis. On the non-Liouville side we prove an abundance principle: the existence of a single minimal positive harmonic function at a prescribed growth scale forces infinitely many. Finally, we clarify the role of moment assumptions in speed theory: any linear-speed law of large numbers on a set of positive probability forces $\mathbb E|X_1|<\infty$, while on torsion-free nilpotent groups one can have $\mathbb E|X_1|=\infty$ yet $|X_n|/n\to0$ in probability.