1 · The essentials
Imagine choosing a network at random in which every point has exactly the same number of connections. The manuscripts study the closely spaced numbers, called eigenvalues, that describe its connection matrix. For any fixed number of connections of at least three, the companion manuscript shows that the local arrangement of these numbers approaches the Gaussian orthogonal ensemble pattern as the network grows. This includes networks with exactly three connections at every point. The main manuscript shows that the same limiting pattern persists when small, independent random changes are added to the matrix’s diagonal. These changes can remain fixed in strength as the network grows, but must be sufficiently weak, and the result concerns locations strictly inside the original spectral band, away from its edges. The main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, think of examining the gaps between nearby fence posts after adjusting your ruler to the local spacing. The manuscripts describe a setting in which those fine-scale arrangements approach the same statistical pattern, both without disorder and with sufficiently weak disorder; this is not a claim that individual eigenvalues remain in place.
Why it matters
The companion manuscript addresses what its abstract calls the fixed-degree bulk-universality conjecture, including degree three. The main manuscript extends this conclusion to sufficiently weak, fixed random diagonal disorder. Together, the results describe a common limiting eigenvalue pattern in both the clean and weakly disordered settings, under explicit restrictions on energy and disorder strength.
2 · Going further
The clean problem concerns the adjacency matrix of a uniformly random simple labelled -regular graph, with a fixed integer . At each fixed energy in the open Kesten–McKay bulk, Manuscript 2 shows that the unfolded eigenvalue point process converges to the GOE bulk process. Its statement covers all admissible graph sizes, without additional conditioning.
Manuscript 1 adds independent, identically distributed uniform diagonal disorder. Its strength is positive, sufficiently small, and fixed as the graph grows. At each fixed energy in a compact subinterval of the clean spectral band, the full microscopic eigenvalue point process is shown in the manuscript to converge to the GOE bulk process after rescaling by the positive density of states of the corresponding infinite-tree operator. The permitted disorder strength depends on and on the distance from the clean spectral edges.
The common formulation is to examine eigenvalues near a fixed energy, rescale their local spacing, and identify a limiting point process. The supplied abstracts do not describe the argument establishing that convergence, so they do not support a more detailed account of the proof strategy. The companion manuscript treats the clean adjacency matrix, while the main manuscript treats weak Anderson disorder; the abstracts do not specify a logical dependency between their arguments.
Useful notions
- Graphs and vertex degree
- Adjacency matrices and eigenvalues
- Independent random variables
- Convergence of random point patterns
- Local spectral density and rescaling
What was known before / what is new
Before
Manuscript 2 identifies fixed-degree bulk universality for random regular graphs as a conjecture that it addresses. The supplied text gives no further account of earlier partial results or their hypotheses.
New
For the clean model, Manuscript 2 shows GOE bulk convergence for every fixed degree and every fixed energy in the open Kesten–McKay bulk, along all admissible graph sizes and without additional conditioning. The main manuscript shows fixed-energy GOE universality with sufficiently weak, positive, fixed iid uniform diagonal disorder. Its conclusion concerns the full microscopic eigenvalue point process, rescaled using the positive density of states of the corresponding infinite-tree operator.
Limits and assumptions
The graphs must be uniformly random, simple, labelled, and regular of fixed degree . In the disordered model, the diagonal variables are independent and identically distributed with a uniform law, and their positive strength must be sufficiently small depending on the degree and distance from the clean spectral edges. The disordered conclusion applies at fixed energies in compact subintervals strictly inside the clean spectral band; no conclusion is supplied for spectral edges, stronger disorder, or other disorder laws. The abstracts do not identify which cases outside this scope remain open. The main result is not formalized in Lean and has not been formally verified.
Mini glossary
- Regular graph —
- A graph in which every vertex has the same number of connections, called its degree.
- Diagonal disorder —
- Random changes to the diagonal entries of the matrix, taken here to be independent and uniformly distributed.
- Spectral bulk —
- The interior region of the spectral band, rather than its edges.
- Microscopic eigenvalue point process —
- The random arrangement of eigenvalues viewed near an energy at the scale of their local spacing.
- Unfolding —
- Rescaling eigenvalues using a local spectral density to compare their fine-scale arrangements.
- GOE bulk process —
- The limiting bulk eigenvalue pattern associated with the Gaussian orthogonal ensemble.
Keywords : Random regular graphs · GOE universality · Anderson disorder
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
For every fixed degree d ≥ 3, the bulk adjacency-eigenvalue point process of a uniform simple random d-regular graph converges to the Gaussian orthogonal ensemble law, including for cubic graphs. The same fixed-energy universality persists under sufficiently weak fixed iid uniform diagonal disorder, throughout compact bands strictly inside the clean spectral edges.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
Fixed-energy universality for weak Anderson disorder on random regular graphs
05/10/2026For every fixed degree d ≥ 3, we prove fixed-energy GOE universality for the adjacency matrix of a uniformly random simple labelled d-regular graph with independent uniform diagonal disorder. The disorder strength is positive, sufficiently small, and fixed as the graph grows. At each fixed energy in a compact subinterval of the clean spectral band, the full microscopic eigenvalue point process converges to the GOE bulk process after rescaling by the positive density of states of the corresponding infinite-tree operator. The permitted disorder strength depends on the degree and the distance from the clean spectral edges.
GOE bulk universality for fixed-degree random regular graphs
23/09/2026We prove the fixed-degree bulk-universality conjecture for random regular graphs. For every fixed integer d ≥ 3 and every fixed energy in the open Kesten–McKay bulk, the unfolded eigenvalue point process of the adjacency matrix of a uniform simple labelled d-regular graph converges to the GOE bulk process. The convergence holds along all admissible graph sizes, without additional conditioning.