Green geometry, Martin boundary and random walk asymptotics on 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.