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Math-2862 · mathematics · Probability

GOE bulk statistics for weakly disordered regular-graph operators

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each fixed regular degree $d\geq 3$, does sufficiently weak fixed iid Anderson disorder on uniform random d-regular graphs yield a positive continuous bulk density and GOE local point-process limits at every energy in the stated bulk interval, after scaling by that limiting density?

Short English statement

For each fixed regular degree $d\geq 3$, does sufficiently weak fixed iid Anderson disorder on uniform random d-regular graphs yield a positive continuous bulk density and GOE local point-process limits at every energy in the stated bulk interval, after scaling by that limiting density?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2862-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2862

For each fixed degree and interior bulk interval, the source allows sufficiently small fixed iid uniform diagonal disorder, proves positive continuous limiting density there, and gives the GOE point-process limit at every fixed energy after unfolding by that density. These are the precise model and limit conventions in the target.

GOE bulk statistics for weakly disordered regular-graph operators

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. The same GOE limit is claimed under small fixed uniform diagonal disorder, unfolded by the corresponding random-tree density of states.

Mathematical statement of the claimed result

Result 1.
Fix $d\ge3$, and take uniform simple labeled $d$-regular graphs, independent iid $\omega_i\sim\operatorname{Unif}[-1,1]$, and $H_{n,d,w}=A_{n,d}+w\operatorname{diag}(\omega_i)$. Let $\nu_{d,w}$ be the expected root spectral measure of the same operator on the infinite $d$-regular tree. For each $0<\kappa<2\sqrt{d-1}$, the source gives $w_0(d,\kappa)>0$. At every fixed $0<w<w_0$, $\nu_{d,w}$ has a continuous positive density $\rho_{d,w}$ near $I=[-2\sqrt{d-1}+\kappa,2\sqrt{d-1}-\kappa]$. For $E\in I$ and every nonnegative $f\in C_c(\mathbb R)$, \[ \left|\mathbb E e^{-\int f\,d\Xi_{n,E}}- \mathbb E e^{-\int f\,d\Xi_n^{\mathrm{GOE}}}\right|\to0, \quad\Xi_{n,E}=\sum_i\delta_{n\rho_{d,w}(E)(\lambda_i-E)}. \] GOE uses variances $1/n$ off diagonal and $2/n$ on diagonal, unfolded at zero by $n/\pi$. Convergence holds through $n>d$ with $nd$ even.

Outline of the argument

Cavity resolvent recursion, edge switches, and spectral averaging yield a distributional local law retaining the tree environment. Conditional entropy identifies Gaussian eigenvector marks without losing second moments; measure-preserving potential changes are compared with Dyson perturbation to obtain fixed-energy universality.

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

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