GOE bulk universality for regular graphs with weak Anderson disorder

The explanation

Take a network of n points where every point has exactly 3 links. Choose it uniformly at random among all such networks. Write its adjacency matrix: a table with a 1 wherever two points are linked. Each row then has exactly three 1s, so the table is almost emptySource 1. The matrix has n eigenvalues, special numbers that summarise how things spread over the network.

For large n, the eigenvalues fill the interval from −2√2 to 2√2, about −2.83 to 2.83. Their density follows the Kesten–McKay formulaSource 2. At the centre, with degree 3, that density is about 0.150. So with a thousand points, neighbouring eigenvalues near zero sit about 0.0067 apart on average. The density gives counts on big intervals. It says nothing about the exact spacings between neighboursSource 3.

The prediction concerns those spacings. Zoom in so the average gap is 1. Physicists expected the gaps to look like those of the Gaussian orthogonal ensemble, or GOE. That is a dense symmetric matrix filled with independent bell-curve numbers. Its eigenvalues repel each other, so very small gaps are rare. Jakobson, Miller, Rivin and Rudnick stated the prediction for fixed degree and gave numerical evidenceSource 4. The striking part is the mismatch in structure. The sparse network looks locally like a tree, yet its spacings should match a dense random matrixSource 5.

Earlier proofs needed the degree to grow. In 2026, Bourgade and Huang proved fixed-energy universality when the degree lies between (log n) to the 24th power and the square root of nSource 6. Picture a million points. Then (log n)^24 is about 2.3 × 10^27, while the square root of n is only 1000. So at that size the range, read literally, is empty. They stated the fixed-degree case as a conjecture.

The first paper here claims to prove that conjecture for every fixed degree from 3 upSource 7. It covers each fixed energy strictly inside the spectrum. It identifies the full pattern of nearby eigenvalues, not just one averageSource 8. No condition such as connectedness is placed on the graph.

The obstacle is that a sparse graph gives little randomness per row. The proof studies eigenvectors, the patterns that go with each eigenvalue. It shows their values at the two ends of a random link behave like Gaussian pairs, to high accuracySource 9. It then uses link swaps, which keep the graph regular, to insert a small Gaussian matrix. A known theorem about Dyson's matrix motion finishes the comparison.

The second paper adds disorder. Each point gets an independent random potential, uniform between −w and w, added on the diagonalSource 10. This is the Anderson model, a basic model of how randomness can trap electrons. The paper claims the same GOE spacings when w is positive, small and fixedSource 11. The allowed size of w depends on the degree and on the distance from the band edgesSource 12. The eigenvalues are rescaled by the density of states of the matching infinite tree.

How sure is this? No Lean formalization is listed for these papers in the openai/math release. So no computer proof checker has checked them. No specialist review has been published.

What it doesn't show. This does not cover the spectral edges, strong disorder, or disorder that depends on the graph size.