Finite-energy lattice percolation with internal critical probability one

Abstract

For every integer d ≥ 2 we construct a bond percolation on Zᵈ that is invariant under all lattice automorphisms, mixing under translations, and has ordinary finite energy. Almost surely it has a unique infinite component, yet every independent Bernoulli thinning with retention p < 1 destroys all infinite components. The construction perturbs a one-ended spanning tree, using the maximum finite-past size along each non-tree edge’s tree path to suppress every bypass of large tree cuts. This answers the historical ordinary-energy question of Benjamini, Häggström and Schramm. We also make explicit a planar threshold-one consequence of Häggström–Mester’s earlier construction. No first-priority claim is made.

The idea in plain English

Percolation studies which connections survive in a random network. This paper constructs a random network on the integer lattice that has one infinite connected component, yet is so fragile that independently deleting each edge with any fixed positive probability destroys every infinite component.

The network has no preferred location or lattice direction. It also has ordinary finite energy: even after all other edges are revealed, any particular edge still has a positive conditional chance of being either present or absent. Those probabilities need not share a common bound away from zero and one.

The construction leaves a sequence of bottlenecks on every route to infinity. Beyond a sufficiently large scale, each such boundary has exactly one surviving crossing edge. Independent thinning eventually breaks these indispensable connections, however close its retention probability is to one.

The result works in every dimension at least two. It develops Häggström and Mester’s earlier one-ended-tree perturbation mechanism and makes a further threshold consequence of their planar example explicit. The note does not claim a verified first resolution of the historical question.

The precise result

For every integer d ≥ 2, there is a bond-percolation law on the nearest-neighbor edges of Zᵈ that is invariant under every graph automorphism and mixing under translations. For each edge e, its conditional probability of being open, given all other output edges, lies strictly between zero and one almost surely.

Almost surely the realized graph X has a unique infinite component C, and p_c(X) = p_c(C) = 1 for independent internal bond thinning. The internal site thresholds of both graphs also equal one. These statements are quenched: X is fixed before the independent thinning is introduced.

More generally, the perturbation applies to any countably infinite connected simple bounded-degree graph with a one-ended spanning tree T. If deleting a tree edge t leaves a finite side of size s_T(t), retain it with probability 1 − 2^(−s_T(t)). Insert a non-tree edge e with probability 2^(−M_T(e)), where M_T(e) is the maximum finite-side size along its connecting tree path. Conditional on T, these choices are independent.

The all-dimensional lattice input comes from Timár’s established factor-of-iid one-ended spanning-tree theorem. The note also derives a planar threshold-one conclusion from the exact fixed-phase midpoint projection of Häggström–Mester’s older model, preserving its original flip rates.

Why the result holds

In a one-ended tree, each edge on a ray separates off a finite set, and those sets grow to exhaust the graph. Any extra edge crossing one of these boundaries has a tree path that includes its separating tree edge.

The insertion rule uses the largest finite-side size along that path. The probabilities of deleting the essential tree edge or adding any bypass have a summable total along the ray. Borel–Cantelli therefore leaves eventual singleton boundaries and retained infinite ray tails.

The retained tails coalesce, yielding one infinite component. Once the graph is fixed, any infinite path must use infinitely many distinct bottleneck edges. Independent retention with p < 1 has probability at most pᵏ of preserving the first k of them, which tends to zero. The same argument uses their vertices for site thinning.

Every conditional edge probability is strictly between zero and one even after the hidden tree is averaged out. Equivariance of the tree input and perturbation preserves all lattice symmetries, while their representation as a measurable factor of iid marks gives translation mixing.

Scope and limitations

The energy condition is ordinary finite energy. The paper neither proves a uniform positive lower bound on the marginal conditional probabilities nor proves that every such uniform bound fails. Small probabilities conditional on the hidden tree do not alone decide the marginal question.

Finite components are allowed. Existence of a unique infinite component does not say that every lattice vertex belongs to it. The external random process is a bond law; the additional site assertion concerns independent thinning inside its realized graph.

The perturbation theorem requires a connected one-ended spanning tree and bounded ambient degree. A disconnected uniform spanning forest in higher dimensions cannot be substituted, since some non-tree edges would have no connecting tree path.

The planar midpoint law is a coordinate marginal of a fixed-phase site construction, with all coarse vertices present. It is not the induced bond graph of the final open sites, and the old site/bond threshold comparison alone does not provide an external finite-energy bond law.

The planar perturbation mechanism is due to Häggström and Mester. The note supplies an additional bounded-cutset deduction and a dimension-independent bypass rule with eventual singleton cuts. Neither first priority for the historical answer nor certified novelty of that rule is claimed. No finite coding radius or quantitative mixing rate is asserted.

Status and AI use

OpenAI tools were used extensively in solving, proof development, drafting, coding, and adversarial verification. The author is responsible for the claims. No completed external human peer review or formal proof-assistant verification is claimed; package checks authenticate files rather than the infinite mathematical proof.

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Collaborate on this result

I welcome substantive collaboration on an independent percolation-theoretic review of the bypass suppression and quenched threshold arguments, the exact relationship to the earlier planar construction, or preparation for journal submission. Expertise in invariant percolation and one-ended spanning trees would be especially useful.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

Cite this paper

Alec Kriebel. (2026). Finite-energy lattice percolation with internal critical probability one (Version 1.0.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23387753

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Background and prior work

• Benjamini, Häggström and Schramm, On the effect of adding ε-Bernoulli percolation to everywhere percolating subgraphs of Zᵈ: Poses the historical ordinary-finite-energy bond question. Its separate sprinkling setting adds fixed-density noise to an everywhere-percolating graph.
• Benjamini and Tassion, Homogenization via sprinkling: Question 1 restates the exact existence problem and defines finite energy as insertion and deletion tolerance. The sprinkling theorem has different hypotheses from this tree-dependent perturbation.
• Häggström and Mester, Some two-dimensional finite energy percolation processes (2009): Section 2 supplies the prior summable finite-past site-flip mechanism. The post-flip threshold-one deduction supplied in this note is not a theorem stated in the inspected antecedent; the original mechanism deserves explicit credit.
• Timár, One-ended spanning trees in amenable unimodular graphs: Theorem 1 provides the equivariant factor-of-iid connected one-ended spanning tree used on every Zᵈ with d ≥ 2.
• Pemantle, Choosing a spanning tree for the integer lattice uniformly: Provides the classical uniform spanning-tree alternative in dimensions two through four and the planar primal/dual input behind the earlier construction.