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Math-1723 · mathematics · Probability

Rough-isometry invariance of percolation nonuniqueness

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

For an infinite connected bounded-degree graph $G$, define Bernoulli bond percolation by independently retaining edges with probability $p$. Let $N_\infty$ count its infinite open components, and set \[ p_c(G)=\inf\{p:\mathbb P_p(N_\infty\ge1)>0\},\qquad p_u(G)=\inf\{p:\mathbb P_p(N_\infty=1)=1\}. \] Both infima are over $[0,1]$; $p=1$ supplies the unique infinite component.

Suppose $G$ and $H$ are infinite connected graphs of bounded degree, and there is a map $f:V(G)\to V(H)$ with constants $A\ge1$, $B\ge0$ satisfying \[ A^{-1}d_G(x,y)-B\le d_H(f(x),f(y))\le A d_G(x,y)+B \] for all $x,y$, with every vertex of $H$ at distance at most $B$ from $f(V(G))$. Prove or disprove \[ p_c(G)<p_u(G)\quad\Longleftrightarrow\quad p_c(H)<p_u(H). \] The rough-isometry terminology here includes multiplicative distortion; the map need not preserve edges, degrees, or a transitive group action. Neither graph is assumed transitive.

On a general graph the set of parameters with a unique infinite cluster need not itself be an interval. Consequently the proposed invariant is precisely the inequality between the two displayed infima, not an unstated monotonicity assertion or a characterization of every intermediate parameter. A counterexample must give a genuine rough-isometry pair and verify opposite truth values of this numerical inequality. The statement concerns invariance of the strict threshold separation, not equality of either individual threshold across the pair.

Short English statement

Is strict separation of percolation and uniqueness thresholds preserved by coarse geometric equivalence of bounded-degree graphs?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1723-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1723

The cited theorem, with amenable transitive uniqueness and rough-isometry invariance of nonamenability, establishes threshold-separation invariance when both graphs are quasi-transitive; our statement allows arbitrary bounded-degree graphs without any transitivity. This entry records partial support; complete coverage of the question is not established by this catalogue.

Rough-isometry invariance of percolation nonuniqueness

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. Every nonamenable locally finite quasi-transitive graph is claimed to satisfy $p_c<p_{2\to2}\leq p_u$, yielding a nonempty phase with infinitely many infinite clusters.

Mathematical statement of the claimed result

Result 1.
Let $G=(V,E)$ have infinitely many vertices, finite degrees, connectedness and quasi-transitivity, with positive vertex Cheeger constant \[ h_V(G)=\inf_{0<|A|<\infty}|\partial_VA|/|A|>0. \] For independent bond percolation, let $T_p(x,y)=\mathbb P_p(x\leftrightarrow y)$, acting on $\ell^2(V)$ with counting measure, and $p_2=\sup\{p:\|T_p\|_{2\to2}<\infty\}$. With $p_c$ and $p_u$ the appearance and uniqueness thresholds, the source claims \[ \sup_{p<p_c}\|T_p\|_{2\to2}=\|T_{p_c}\|_{2\to2}<\infty, \qquad p_c<p_2\le p_u. \] It further supplies $p_c<p_1<p_2'<1$ such that the uniform-edge-label coupling almost surely has infinitely many infinite clusters simultaneously at every $p\in[p_1,p_2']$.

Outline of the argument

For the new unimodular case, local endpoint maps control moments of finite critical clusters. A deformation lowers susceptibility while retaining a large connectivity-operator norm, thereby proving the strict operator-threshold separation; the nonunimodular case uses the established operator result.

Formal verification

For family 214, the documented statements include the strict operator-threshold separation, the nonunique percolation interval, and selected critical moment and tail bounds, with a Cayley-graph specialization. The listed Comparator files are BenjaminiSchramm.lean, CayleyPercolation.lean. The inventory does not supply a named declaration for these Comparator entries. 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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