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Math-2895 · mathematics · Probability

Finite critical two-point operator norm and nonuniqueness for nonamenable graphs

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every infinite connected locally finite nonamenable quasi-transitive graph, does independent bond percolation satisfy $p_c<p_{2\to 2}\leq p_u$ and have a bounded l2 two-point connectivity operator at $p_c$? Here $p_{2\to 2}$ is the supremum of p for which that operator is bounded, and $p_u$ is the uniqueness threshold. Include the stated simultaneous nonuniqueness interval for two parameter configurations.

Short English statement

For every infinite connected locally finite nonamenable quasi-transitive graph, does independent bond percolation satisfy $p_c<p_{2\to 2}\leq p_u$ and have a bounded l2 two-point connectivity operator at $p_c$? Here $p_{2\to 2}$ is the supremum of p for which that operator is bounded, and $p_u$ is the uniqueness threshold. Include the stated simultaneous nonuniqueness interval for two parameter configurations.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2895-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2895

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. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Finite critical two-point operator norm and nonuniqueness for nonamenable graphs

OpenAI's claim

OpenAI presents a proof 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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