Gaussian-matrix statistics inside sparse random graphs

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Family 219 · Probability and statistical mechanics
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-11

1. The question

A large random regular graph is extremely sparse: every vertex has exactly the same fixed number of neighbors, even as the graph grows. A Gaussian random matrix is dense, with randomness spread across essentially all entries. Why should their eigenvalues have the same fine-scale spacing patterns?

The distinction between a broad spectral histogram and microscopic spacings is crucial. Two ensembles can have very different overall densities while showing the same local repulsion after distances are measured in units of the average spacing.

This family’s two manuscripts claim that phenomenon for every fixed graph degree at least three. One treats the ordinary adjacency matrix. The other adds a small, fixed random potential at each vertex, obtaining the same microscopic law in the interior of the original spectral band. The limit is the bulk eigenvalue process of the Gaussian orthogonal ensemble, or GOE.

2. Background and essential definitions

A graph is d-regular if each vertex has exactly $d$ neighbors. Choose a graph uniformly among simple labelled $d$-regular graphs on $n$ vertices. “Simple” excludes loops and multiple edges; “labelled” distinguishes the vertices. Sizes must be admissible, in particular $nd$ must be even.

The adjacency matrix $A_n$ has entry one when two vertices are joined and zero otherwise. It is real symmetric, so its eigenvalues $\lambda_1,\ldots,\lambda_n$ are real. We use ordinary adjacency, without dividing by $\sqrt{d-1}$.

The limiting density for most eigenvalues is the Kesten–McKay density

$$\rho_d(E)= \frac{d\sqrt{4(d-1)-E^2}}{2\pi(d^2-E^2)}, \qquad |E|<2\sqrt{d-1}.$$

The open interval in this formula is the spectral bulk. The familiar eigenvalue $d$, coming from the constant vector, lies outside it and does not affect an interior microscopic window.

Near a bulk energy $E$, a short interval of length $\Delta$ contains roughly $n\rho_d(E)\Delta$ eigenvalues. The average spacing is therefore about $1/(n\rho_d(E))$. Unfolding rescales the eigenvalues by that local density:

$$\Xi_{n,E}=\sum_{i=1}^n \delta_{\,n\rho_d(E)(\lambda_i-E)}.$$

Here $\delta_x$ denotes a unit point at $x$. The random measure $\Xi_{n,E}$ records all nearby eigenvalues together, with average density one in the limit.

The normalized GOE is a real symmetric matrix whose independent upper-triangular entries are Gaussian, with variance $1/n$ off the diagonal and $2/n$ on it. Its limiting density at zero is $1/\pi$, so its unit-density reference process uses the scaling $(n/\pi)\mu_i$, where $\mu_i$ are its eigenvalues.

For the disordered model, put

$$H_n=A_n+w\operatorname{diag}(\omega_1,\ldots,\omega_n),$$

where the $\omega_v$ are independent uniform variables on $[-1,1]$, independent of the graph. The diagonal term is a random on-site potential. This is a finite-graph Anderson model.

3. Development history and previous partial results

Kesten’s work on regular trees and McKay’s random-graph calculation identified the macroscopic density. Jakobson, Miller, Rivin, and Rudnick later formulated the GOE spacing prediction for fixed-degree regular graphs and supplied numerical evidence. Knowing the density does not settle the prediction: equally spaced points, independent random points, and repelling points can all have the same average density. McKay’s density theorem, Eigenvalue spacings for regular graphs

Local laws brought finer control. They estimate resolvents and eigenvalue counts in shrinking windows. Bauerschmidt, Knowles, Huang, and Yau developed these estimates and proved bulk universality in ranges where the degree grows with graph size. Huang and Yau established fixed-degree spectral control for every $d\geq3$. Huang, McKenzie, and Yau obtained optimal rigidity and GOE edge universality. Edge statistics describe extreme nontrivial eigenvalues, a different regime from a fixed interior energy. Bulk statistics at growing degree, Fixed-degree edge universality

The clean manuscript also credits Bourgade and Huang’s 2026 fixed-energy result for a degree range starting above a power of $\log n$. Its remaining fixed-degree conjecture is the target here. The important advance is therefore both fixed degree and fixed energy, rather than an average over a moving spectral window. Bourgade–Huang

Eigenvectors provide another ingredient. Backhausz and Szegedy used entropy inequalities to relate almost-eigenvectors of random regular graphs to Gaussian waves on the infinite tree. Their qualitative weak limits could lose some variance. The new argument needs quantitative moment control that preserves the full variance at microscopic energies. Almost eigenvectors

For Anderson disorder, Klein proved extended spectral behavior on the infinite regular tree at weak disorder in compact subsets of the clean band. Later work refined stability and established quantum ergodicity on suitable finite graphs. These statements concern spectral type or spatial equidistribution, rather than the joint spacing process of finite-graph eigenvalues. M1 addresses that additional microscopic question. Klein’s extended-state theorem

4. What the new manuscripts claim

Clean fixed-degree graphs

M2 states that for every fixed integer $d\geq3$ and every fixed energy

$$-2\sqrt{d-1}<E<2\sqrt{d-1},$$

the unfolded point process $\Xi_{n,E}$ converges to the GOE bulk process along all admissible graph sizes. No conditioning on connectedness, nonbipartiteness, or another graph property is imposed.

The theorem is expressed through Laplace functionals. For every nonnegative continuous compactly supported test function $f$, it compares

$$\mathbb E\exp\!\left(-\int f\,d\Xi_{n,E}\right)$$

with the corresponding GOE quantity. Since the integral is a sum of $f$ over the local eigenvalues, these tests encode the entire point-process law. The conclusion is stronger than a mean density or one spacing statistic.

Fixed weak diagonal disorder

M1 fixes a distance $\kappa>0$ from the clean spectral edges and considers

$$I_{d,\kappa}= [-2\sqrt{d-1}+\kappa,\,2\sqrt{d-1}-\kappa].$$

For sufficiently small fixed $w>0$, depending on $d$ and $\kappa$, the infinite-tree Anderson operator has a continuous, strictly positive density of states $\rho_{d,w}$ on this interval. Its measure is defined by averaging the root spectral measure over the independent tree potentials.

For each fixed $E\in I_{d,\kappa}$, M1 claims GOE convergence after unfolding with

$$n\rho_{d,w}(E)(\lambda_i-E).$$

The density is that of the disordered infinite-tree operator, not the clean Kesten–McKay formula. The strength $w$ remains positive and fixed as $n\to\infty$. The degree, energy, and test function are likewise fixed first.

The stated eigenvalue convergence averages over both graph and potential randomness. An environment-conditioned Gaussian statement for eigenvector marks is an intermediate ingredient; it should not be confused with a separate quenched eigenvalue-universality theorem for every fixed potential realization.

5. Why the advance matters

Importance: 4/5. The clean claim resolves the fixed-degree bulk-universality problem, including degree three, where the graph remains sparse at every size. The extension with a fixed weak random potential connects that universality to a basic model of disordered quantum transport.

The result would show that local tree geometry and GOE eigenvalue repulsion are compatible. Tree structure determines the local density and eigenvector covariance, while the global finite graph supports the same microscopic spectral statistics as a dense Gaussian matrix.

Its reach is substantial, but the theorem remains in the bulk and, for the Anderson model, in a specified weak-disorder regime. It does not describe the entire localization phase diagram.

6. The main idea

Use the tree as a local reference

Most fixed-radius neighborhoods of a random regular graph look like trees when $n$ is large. For a matrix $H$, the resolvent $(H-z)^{-1}$, with $\operatorname{Im}z>0$, records smoothed spectral information.

On a tree, deleting the edge to a parent leaves independent child branches. The diagonal branch resolvent satisfies

$$g(z)=\left(V-z-\sum_{j=1}^{d-1}g_j(z)\right)^{-1},$$

where $V$ is the root potential and the $g_j$ are branch values. In the clean case this reduces to a deterministic equation. With disorder, the limiting resolvent remains random, even though the averaged density of states is deterministic.

M1 controls this random recursion near the clean bulk and compares finite-graph cavity resolvents with it. Replacing the random tree response by only its mean would lose information needed later.

Recover Gaussian eigenvector marks

Normalize an eigenvector so its squared coordinate sum is $n$. Its value at a uniformly sampled vertex then has average square one. In the clean model, the manuscript obtains joint Gaussian moment approximations for finitely many eigenvectors sampled at graph-edge endpoints. The two endpoints have covariance $E/d$, reflecting their adjacency.

The argument combines counts of local colored configurations with entropy and Fisher-information inequalities. Gaussian smoothing turns the counting constraint into an analytic comparison. Quantitative control of the equality case forces Gaussian moments and prevents variance from escaping into rare large coordinates.

With disorder, the marks are Gaussian conditional on the tree environment, with variance proportional to the imaginary part of the local tree resolvent. That variance varies from place to place. Keeping it conditional is part of the extension’s substance.

Insert Gaussian noise without changing the ensemble

A degree-preserving edge switch changes only a small part of a regular graph and can be chosen reversibly, preserving its distribution. In the disordered model, suitable exchanges of intervals in the potential law play a related role.

The effect on microscopic eigenvalues can be written using the Gaussian marks. Many weak changes then approximate an additive GOE perturbation. The number of changes is increased inside an already constructed limiting marked process, after the graph-size limit; the method does not assume a growing-dimensional eigenvector theorem for finite graphs.

Finally, fixed-energy Dyson Brownian motion results identify the local statistics of the Gaussian-perturbed matrices. The manuscripts match the density normalization and control eigenvalue counts so that the conclusion concerns the full point process. Fixed-energy Dyson Brownian motion

7. Scope, limitations, and open questions

The degree is a fixed integer at least three. Degree two gives unions of cycles and is outside both claims. The energy stays in the open clean bulk; the disordered theorem keeps a prescribed positive distance from its edges. No conclusion is asserted here for edge fluctuations, spectral outliers, or energies drifting toward a band boundary with $n$.

The random potential is independent and uniform on a bounded interval. The theorem does not automatically extend to arbitrary potential distributions, correlated disorder, or disorder strong enough to produce localization. Its smallness threshold depends on the degree and the chosen interior energy range.

Universality of eigenvalue statistics does not mean that eigenvectors have constant local variance in a disordered environment. Nor does it by itself prove a transport theorem on every finite or infinite graph. The density, local tree response, and point-process limit describe different aspects of the model.

Natural further questions include the boundary of the GOE regime as disorder grows, other potential laws, and the transition to localized spectral statistics. These require information beyond the compact weak-disorder bulk treated here.

8. Sources and further reading

M2 develops the clean fixed-degree theorem and the entropy-to-switching mechanism. M1 adapts it to random tree resolvents and positive fixed diagonal disorder.

For earlier milestones, see McKay on the limiting density, Backhausz–Szegedy on Gaussian-wave eigenvectors, and Huang–McKenzie–Yau on fixed-degree edges. The final analytic comparison uses Landon–Sosoe–Yau’s fixed-energy universality theorem.

Source manuscripts

• M1. OpenAI, Fixed-energy universality for weak Anderson disorder on random regular graphs (October 5, 2026).
• M2. OpenAI, GOE bulk universality for fixed-degree random regular graphs (September 23, 2026).

Sources consulted on 11 October 2026 at repository revision fd4aeeb. Download the manuscript citations.