Weighted isoperimetry implies percolation

This paper is a theoretical contribution to probability theory (percolation), proving that a weighted isoperimetric inequality on an edge-weighted graph implies the existence of an infinite open cluster (percolation) under independent bond percolation with edge-weight-dependent probabilities. It extends the recent resolution of the Benjamini–Schramm conjecture (the "p_c < 1 question" for graphs of isoperimetric dimension d > 1) from the homogeneous-weight case to the fully inhomogeneous setting, and leverages this generality to settle two independent long-standing conjectures.

Core Contribution. The central novelty is a new Peierls-type argument built around the notion of *cohesion*. Prior approaches (Easo–Severo–Tassion 2025; the AI-generated proofs Cut26a/b) reduce percolation to counting minimal cutsets and constructing a random cutset via a Karger-style contraction algorithm, then bounding the probability that a cutset is ω-closed. The authors correctly identify that this cutset-counting strategy *necessarily fails* in the weighted regime because as edge weights vanish (λ→0), small local cutsets proliferate and the union bound diverges — this is exactly the regime relevant to both target applications (truncation with Λ_M = o(log M); transitive graphs with λ = C/Δ, Δ→∞). Their fix is to define ω-cohesive cutsets — closed cutsets whose two sides additionally contain no small (weight < 1) uncrossed cut — and to weight the contraction algorithm by edge weights, coupling it to the percolation via a Poissonization/monotonicity trick reminiscent of Kruskal's MST algorithm. The key inequality (5) accounts for internal/external connectivity costs beyond the blocking surface. This is a genuinely new conceptual ingredient, not a mechanical generalization.

Applications. (1) It proves that every non-summable, non-collinear translation-invariant long-range percolation on ℤ^d (d≥2) admits a percolating truncation — resolving the 1999 Sidoravicius–Surgailis–Vares conjecture and its Friedli–de Lima generalization, which had only been established under progressively weaker but still restrictive symmetry/dimension assumptions over ~25 years. (2) It establishes a universal bound p_c ≤ C/Δ for all transitive graphs of superlinear growth and degree Δ, confirming Easo–Hutchcroft Conjecture 7.3, and as a corollary settles the previously-open gap-at-1 for *site* percolation on transitive graphs. These are both notable, well-known open problems, giving the paper unusually high external validation.

Methodological Rigor. As pure mathematics, the paper's evidence is its proofs, which are careful, self-contained (an appendix reproduces the isoperimetric bookkeeping lemma), and structured logically from the main theorem through the two application-specific isoperimetric lemmas. The reduction of the applications to short isoperimetric inequalities (Lemma 3.1 via projection/folklore arguments; Lemma 4.1 via Tessera–Tointon geometric group theory) is clean and the optimality remark (4.2) shows the authors understand the sharpness of their bounds. The constants are explicit and tracked throughout. I see no evident gaps; the infinite-graph case is reduced rigorously to the finite case by wiring.

Timeliness & Relevance. Extremely timely. It directly builds on 2025–2026 work, including the striking fact that the homogeneous Benjamini–Schramm case was recently settled by an AI system (Cut26a/b). The paper explicitly notes those results do *not* suffice for the applications — positioning this work as the necessary human conceptual advance that makes the machinery applicable to the hard, weighted cases. This is a live, fast-moving frontier.

Strengths. (i) Solves multiple independent, aged conjectures with a single unified theorem — a hallmark of impactful mathematics. (ii) The cohesion idea is a reusable primitive likely to influence future work on percolation via isoperimetry, giant components, and connectivity thresholds (the authors themselves sketch applications to giants and full connectivity in Remark 1.5). (iii) The theorem gives quantitative surface-order exponential decay and explicit constants, increasing its utility. (iv) Broad scope: finite and infinite graphs, general weighted isoperimetric profiles.

Limitations. (i) The constants (e.g., C = 1/625, the universal C in Theorem 1.7) are far from optimal, as the authors acknowledge — so the result is qualitative rather than sharp in the applications. (ii) The work is intrinsically within the percolation/statistical-mechanics community; interdisciplinary reach is modest (probability + combinatorics + geometric group theory as an input). (iii) It has essentially no near-term applied/translational component — this is foundational pure math. (iv) The novelty, while real, sits atop a very recent scaffold (the ESZ and AI cutset-counting framework); the contribution is the cohesion refinement rather than an entirely new paradigm.

Overall. This is a strong, well-executed theory paper that resolves genuinely hard, long-standing conjectures via a new and likely-reusable argument. Within its subfield it is a clear standout; its ceiling is limited only by the narrowness of the audience and the non-sharp constants. The results and conclusions are not especially surprising (the conjectures were widely believed), which slightly tempers its disruptive potential, but the method's cleanliness and the breadth of what it unifies mark it as an above-average, influential contribution.