Green geometry, Martin boundary and random walk asymptotics on groups
Mayukh Mukherjee, Soumyadeb Samanta, Soumyadip Thandar
TL;DR
The paper introduces the Green-variation functional $\Delta(S;a,b)$ as a computable certificate linking Martin boundary collapse, Green geometry, and random-walk speed on finitely generated groups. It proves that $\Delta\to0$ along exhaustions is equivalent to the strong Liouville property, and provides two general, checkable mechanisms to verify $\Delta$-decay: heat-kernel envelopes with Tauberian comparability and an elliptic Hölder exhaustion criterion. An obstruction result shows that balls cannot realise $\Delta\to0$ on groups of exponential growth under mild on-diagonal bounds, delineating the non-Liouville side. Consequences include linear-scale Green-geometry collapse and vanishing Green speed under boundary collapse, and an abundance phenomenon for non-liouville settings: the existence of a single minimal harmonic function at a growth scale implies infinitely many. The work further clarifies how moment assumptions influence speed, showing that genuine linear-speed laws enforce finite first moments, while heavy-tailed constructions can yield infinite first moment with vanishing speed in probability on nilpotent and related 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 \[ Δ(S;a,b):=\max_{x\in\partial S}\frac{|G(a,x)-G(b,x)|}{G(a,x)}. \] We prove that $Δ\to0$ 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, $Δ$ 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.