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Math-1721 · mathematics · Probability

Uniqueness of percolation on graphs roughly isometric to lattices

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Fix $d\ge1$. Let $G$ be an infinite connected graph of uniformly bounded degree which is roughly isometric to the nearest-neighbor lattice $\mathbb Z^d$. Explicitly, suppose there are constants $A\ge1$, $B\ge0$ and a map $f:V(G)\to\mathbb Z^d$ such that \[ A^{-1}d_G(x,y)-B\le\|f(x)-f(y)\|_1\le A d_G(x,y)+B \] for all vertices $x,y$, and every lattice point lies within $\ell_1$-distance $B$ of some image point. This is the source's multiplicative-and-additive rough-isometry convention.

For a fixed $p\in[0,1]$, independently declare each edge of $G$ open with probability $p$. Let $N_\infty$ be the number of infinite connected components of the open-edge subgraph. Must \[ \mathbb P_p(N_\infty\le1)=1 \] hold for every such $G$ and every $p$?

The assertion includes parameters where no infinite component exists. When percolation does occur, it asks for uniqueness throughout the entire parameter range, including any critical value. No invariance of $G$ under a transitive group, and no translation-invariant choice of the coarse comparison map, is assumed. These omissions are central because ordinary lattice uniqueness arguments use spatial symmetries absent from an arbitrary roughly isometric graph.

The degree bound is uniform over the vertices of each graph, although it and the rough-isometry constants may depend on $G$. The problem concerns Bernoulli bond percolation with one constant parameter, rather than arbitrary dependent or spatially inhomogeneous random subgraphs.

Short English statement

Does every bounded-degree graph with the large-scale geometry of a Euclidean lattice have at most one infinite Bernoulli cluster?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1721-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1721

The cited theorem gives zero infinite clusters at pc<1 on the quasi-transitive members of our coarse-lattice class; our question quantifies every parameter and allows arbitrary bounded-degree nontransitive roughly-isometric graphs. This entry records partial support; complete coverage of the question is not established by this catalogue.

Uniqueness of percolation on graphs roughly isometric to lattices

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. Critical bond percolation is claimed to have no infinite component on every infinite locally finite quasi-transitive graph with $p_c<1$.

Mathematical statement of the claimed result

Result 1.
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

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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