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Math-1735 · mathematics · Probability

Uniqueness at the percolation uniqueness threshold

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Let $G$ be an infinite connected locally finite quasi-transitive graph, meaning that its automorphism group has finitely many orbits on vertices. In Bernoulli site percolation, retain each vertex independently with probability $p$ and consider the subgraph induced by the retained vertices. Denote its number of infinite connected components by $N_\infty(p)$, and put \[ p_u(G)=\inf\{p\in[0,1]:\mathbb P_p(N_\infty(p)=1)=1\}. \] Characterize the graphs in this class for which \[ \mathbb P_{p_u(G)}(N_\infty(p_u(G))=1)=1. \] The desired characterization should give necessary and sufficient structural conditions on the underlying graph, rather than restating the probability-one condition as a definition. The same classification question can be posed for Bernoulli bond percolation, with independent open edges.

This is specifically an endpoint question. The original list also asked for uniqueness for every $p>p_u$; its later update records that part as resolved. Uniqueness above the infimum does not determine whether it holds at the infimum itself. At the endpoint the alternative to exactly one infinite component can be absence of infinite components or the presence of infinitely many; a characterization must accommodate those distinct failures.

No planarity, amenability, Cayley-graph representation, or fixed degree is prescribed beyond local finiteness and quasi-transitivity. When $p_u=1$, site percolation retains the entire connected graph at the threshold, yielding a basic class with uniqueness. The difficult classification includes thresholds strictly below one and the effect of the graph's large-scale structure on critical uniqueness.

Short English statement

Which quasi-transitive graphs have a unique infinite percolation cluster exactly at the uniqueness threshold?

Sources

  • S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 10000088 (AMR-099-0088), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 Benjamini and Schramm - Percolation beyond Z^d with 1999 update, Question 5, PDF page 10; at-threshold remainder, PDF page 10. Original source indexed by AMR-099-0088. https://emis.de/ft/41630#page=10.
  • S3 I. Benjamini and O. Schramm, Percolation beyond Z^d, many questions and a few answers, Questions 3 and 5; site-percolation convention in Section 2. https://emis.de/ft/41630.
  • S4 T. Hutchcroft and M. Pan, Percolation at the uniqueness threshold via subgroup relativization (2024), partial results for bond percolation on groups. https://arxiv.org/abs/2409.12283.

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1735-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1735

For the lattice Z3, whose Bernoulli uniqueness threshold equals its critical threshold by amenable-lattice uniqueness, The cited theorem shows absence of a cluster at that endpoint; it supplies one negative case, not necessary-and-sufficient conditions for all quasi-transitive graphs. The cited theorem rules out bond clusters at pc<1; for amenable quasi-transitive graphs with pc=pu by the usual amenable uniqueness theorem this gives negative endpoint cases, but no full site/bond structural characterization at pu. This entry records partial support; complete coverage of the question is not established by this catalogue.

Uniqueness at the percolation uniqueness threshold

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: Both bond and site percolation on $\mathbb Z^3$ are claimed to have only finite clusters at their respective critical probabilities; Critical bond percolation is claimed to have no infinite component on every infinite locally finite quasi-transitive graph with $p_c<1$. Together they provide the following support within the scopes described below.

Mathematical statement of the claimed result

Result 1.
For independent nearest-neighbor bond percolation on $\mathbb Z^3$, write $p_c^{\mathrm{bond}}$ for the first parameter permitting an infinite cluster. The source asserts \[ \mathbb P_{p_c^{\mathrm{bond}}}(\exists\text{ infinite open cluster})=0. \] For independent nearest-neighbor site percolation, it separately claims \[ \mathbb P_{p_c^{\mathrm{site}}}(\exists\text{ infinite open cluster})=0. \] These are two cubic-lattice statements at their respective critical parameters.
Result 2.
Suppose $G$ is infinite, connected, locally finite, undirected, and has finitely many vertex orbits under automorphisms. For Bernoulli bond percolation, define $p_c(G)$ as the infimum of parameters with positive probability of an infinite cluster. The claimed implication is \[ p_c(G)<1\quad\Longrightarrow\quad \mathbb P_{p_c(G)}(\exists\text{ infinite open cluster})=0. \] Equivalently, every fixed vertex $x$ satisfies $\mathbb P_{p_c}(|C_x|=\infty)=0$. The hypothesis $p_c<1$ is essential to the stated scope.

Outline of the argument

Independent hyperedges provide a finite connection inequality that accommodates the two models. Finite-scale extension bounds allow an adaptive exploration to maintain successful connections; assuming critical percolation would make this construction survive below the threshold, a contradiction.

The proof separates growth regimes and uses finitary structural reduction in the polynomial-growth case. Two coordinate moves are glued through fresh corridors using a conditional connection inequality; adaptive exploration would then produce subcritical percolation if an infinite critical cluster existed.

Formal verification

For family 213, the Lean scope covers cubic-lattice bond and site critical nonpercolation, and quasi-transitive bond graphs when $p_c<1$. The listed Comparator files are CriticalPercolation.lean, CriticalZ3.lean. Named declarations include OAI.CriticalZ3.critical_no_infinite. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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