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Math-2412 · mathematics · Probability

Eigenvalue Counts for Low-Degree Random Regular Graphs

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For uniformly random $d$-regular graphs on $n$ vertices, improve the estimates for the number of adjacency eigenvalues in a specified interval in the low-degree regime $d\lesssim(\log n)^4$. The target is a quantitative eigenvalue-counting estimate in that regime.

Formulation references: [S1], [S2].

Short English statement

For uniformly random $d$-regular graphs on $n$ vertices, improve the estimates for the number of adjacency eigenvalues in a specified interval in the low-degree regime $d\lesssim(\log n)^4$. The target is a quantitative eigenvalue-counting estimate in that regime.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2412-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2412

The question asks for improved quantitative bounds on interval eigenvalue counts when the degree is at most logarithmic to the fourth power. The selected theorem describes the limiting bulk point process for each fixed degree and fixed interior energy. It consequently identifies limiting microscopic count distributions, but gives neither a convergence rate nor a uniform estimate in the requested degree range. Its auxiliary local counting estimates are imported inputs. This is a related universality result rather than a quantitative solution of the recorded counting problem.

Eigenvalue Counts for Low-Degree Random Regular Graphs

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. Every fixed regular degree $d\geq3$ is claimed to give GOE eigenvalue statistics at fixed interior bulk energies for uniformly sampled simple regular graphs.

Mathematical statement of the claimed result

Result 1.
For fixed $d\ge3$, let $G_{n,d}$ be uniform among simple labeled $d$-regular graphs, at admissible sizes, and $\lambda_i$ its adjacency eigenvalues. Fix $|E|<2\sqrt{d-1}$ and define \[ \rho_d(E)=\frac{d\sqrt{4(d-1)-E^2}}{2\pi(d^2-E^2)},\qquad \Xi_{n,d,E}=\sum_i\delta_{n\rho_d(E)(\lambda_i-E)}. \] For GOE eigenvalues $\mu_i$, with off-diagonal variance $1/n$ and diagonal variance $2/n$, set $\Xi_n^{\mathrm{GOE}}=\sum_i\delta_{(n/\pi)\mu_i}$. The claim is that each nonnegative continuous compactly supported $h$ satisfies \[ \left|\mathbb E e^{-\int h\,d\Xi_{n,d,E}} -\mathbb E e^{-\int h\,d\Xi_n^{\mathrm{GOE}}}\right|\to0. \] The random graph has no further conditioning.

Outline of the argument

An entropy method gives quantitatively accurate Gaussian eigenvector marks and high moments at sampled edges. Reversible edge switches promote those marks to microscopic orthogonal invariance; a Gaussian perturbation and fixed-energy Dyson relaxation identify the point process, with compact-count control completing convergence.

Formal verification

Family 219 has no Lean scope page in this release. Its selected result is not presented as formalized. 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.

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