Continuity for random walks in hyperbolic percolation: drift, entropy, and dimension

Impact Assessment

Core Contribution. This paper proves that for simple random walk on infinite supercritical Bernoulli percolation clusters on Cayley graphs of non-elementary hyperbolic groups, the drift ℓ(p) and asymptotic entropy h(p) depend continuously on the percolation parameter p throughout the supercritical phase (p_c, 1]. Combined with the author's earlier dimension formula (Sak24), this yields continuity of the Hausdorff dimension of harmonic measure on the Gromov boundary. The result answers a natural regularity question implicit in the foundational Benjamini–Lyons–Schramm program (BLS99), which established positivity of these quantities but left their dependence on p unexplored. A notable feature is that continuity is proven *across the uniqueness threshold p_u*, where p_c < p_u < 1 for one-ended hyperbolic graphs — a regime where geometric quantities (limit-set dimension) are known to possibly jump (HP24), making the smoothness of the *measure-theoretic* quantity a genuine and non-obvious distinction.

Methodological Rigor. The paper is a pure-mathematics contribution and the arguments appear complete and careful. The proof strategy is well-motivated and clearly explained. The central technical innovation is the notion of "stationary families" — measurable families of boundary measures satisfying a rerooting relation — which allows measures at different parameters to live in a common compact space, so that compactness and weak-limit arguments can transfer stationarity across p (Lemma 3.4). Two components stand out: (1) a uniqueness theorem for the stationary family on the Gromov boundary (Theorem 3.8), proven via a bilateral walk and boundary contraction outside neighborhoods of the backward limit, adapting the classical fixed-measure uniqueness argument to the environment-dependent setting where rerooting changes the boundary measure; and (2) a random-environment version of the Kaimanovich–Vershik entropy formula (Theorem 5.5), combined with lower semicontinuity of KL divergence following Silva's recent softer approach (Sil26). The drift argument adapts Erschler–Kaimanovich via a Furstenberg-type formula on the horoboundary. The reliance on existing scaffolding is honestly acknowledged.

Potential Impact. The work sits in a specialized but active subfield at the intersection of probability, ergodic theory, and geometric group theory. The regularity of random-walk characteristics has attracted sustained attention (Ledrappier, Mathieu, Gouëzel, Erschler–Kaimanovich, Mathieu–Sisto, Choi, Silva), but almost all prior work perturbs the *driving measure* on a fixed graph; perturbing the *environment* via a percolation parameter is a comparatively fresh direction. The "stationary families" formalism and the environment-dependent uniqueness/KV arguments are reusable primitives that could support future work on stronger regularity or on monotonicity (the long-standing BLS Conjecture 4.10 on entropy monotonicity remains open). The explicit motivation from Gu–Zhao's C^∞ diffusivity result on Z^d frames continuity as a first step toward potentially much stronger regularity, which the paper's own Questions 1.3–1.4 flag.

Timeliness & Relevance. The paper is timely: it engages directly with 2024–2026 work (Hutchcroft–Pan, Gu–Zhao, Silva, Choi, Geldbach) and builds on Hutchcroft's 2019 p_c < p_u theorem. The question is natural and the subfield is presently active, so the contribution lands in a receptive context.

Strengths. (1) A clean, complete answer to a well-posed question. (2) A genuinely new technical framework (stationary families) adapted to random environments. (3) The uniqueness-on-boundary argument is elegant and non-trivial. (4) Clear exposition with a well-structured proof outline. (5) Broad applicability within its class — the result holds for *any* Cayley graph of *any* non-elementary hyperbolic group.

Limitations. (1) Continuity is a relatively *weak* form of regularity; the more consequential and interesting questions (differentiability, analyticity, monotonicity) are left entirely open, and the Gu–Zhao comparison implicitly highlights how much further one might go. (2) The result is confined to hyperbolic groups, where strong boundary theory is available; extension beyond this setting is unclear. (3) Much of the machinery (Furstenberg formula, KV formula, KL lower semicontinuity, boundary contraction) is adapted rather than newly invented — the novelty lies in the environment-dependent adaptation, not in fundamentally new tools. (4) As a purely theoretical result, it has no direct applications outside mathematics.

Other Observations. The paper is self-contained and reproducible at the level appropriate for pure math — all constructions and proofs are spelled out. It corroborates and extends the intuition that measure-theoretic boundary quantities behave more regularly than geometric ones, a theme it thoughtfully contextualizes via the branching-Brownian-motion analogy (Lalley–Sellke, Geldbach). The result does not contest or overturn any prior claim; it fills a gap. Its likely trajectory is to be cited as a useful reference within the random-walks-in-random-environments and boundary-theory communities, and possibly built upon in efforts to strengthen the regularity or attack the entropy-monotonicity conjecture, rather than to reshape the field broadly.

Overall, this is a solid, technically competent contribution that meaningfully advances a niche question and introduces a reusable conceptual framework, but whose immediate influence is bounded by the modesty of the "continuity" conclusion and the specialization of its setting.