Green geometry, Martin boundary and random walk asymptotics on groups

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 $$\mathrm{\Delta }(S;a,b):=\underset{x\in \mathrm{\partial }S}{\max }\frac{\mathrm{\mid }G(a,x)-G(b,x)\mathrm{\mid }}{G(a,x)}\mathrm{.}$$ We prove that $\mathrm{\Delta }\to 0$ along exhaustions characterises the strong Liouville property (under mild, verifiable hypotheses on the strong Liouville $\Rightarrow Δ\to0$'' direction), turning boundary oscillation estimates for Green kernels into potential-theoretic rigidity. We then give two general criteria for $Δ$-vanishing. The first one derives quantitative bounds on $Δ$ 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 Hölder exhaustion’’ criterion. Conversely, on groups of exponential growth, $\mathrm{\Delta }$ 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 ($d_G(e,x)=o(\mathrm{\mid }x\mathrm{\mid })$) 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 $\mathbb{E}\mathrm{\mid }{X}_1\mathrm{\mid }<\mathrm{\infty }$, while on torsion-free nilpotent groups one can have $\mathbb{E}\mathrm{\mid }{X}_1\mathrm{\mid }=\mathrm{\infty }$ yet $\mathrm{\mid }{X}_n\mathrm{\mid }\mathrm{/}n\to 0$ in probability.