Ramanujan Property and Edge Universality of Random Regular Graphs

TL;DR

The paper analyzes the spectrum of the normalized adjacency matrix $H=A/\\sqrt{d-1}$ for random $d$-regular graphs, proving optimal eigenvalue rigidity and edge universality. By combining loop equations with a self-consistent Green’s-function approach and a novel local-resampling scheme, the authors derive microscopic loop-equation structure at the spectral edge and establish that edge fluctuations converge to the Tracy–Widom$_1$ distribution from GOE. They show that a positive fraction (about 69%) of large $d$-regular graphs are Ramanujan, with the second-largest and smallest nontrivial eigenvalues tightly controlled. The methodology hinges on a new Woodbury-based expansion for Green’s function differences, an iterative resampling framework organized by forests of switching edges, and careful error bookkeeping that yields optimal rigidity and universal edge statistics in the fixed-degree regime. Overall, the work provides a rigorous bridge between random-regular-graph spectra and universal random-matrix edge behavior, with broad implications for expander construction and spectral graph theory.

Abstract

We consider the normalized adjacency matrix of a random $d$-regular graph on $N$ vertices with any fixed degree $d\geq 3$ and denote its eigenvalues as $λ_1=d/\sqrt{d-1}\geq λ_2\geqλ_3\cdots\geq λ_N$. We establish the following two results as $N\rightarrow \infty$. (i) With high probability, all eigenvalues are optimally rigid, up to an additional $N^{{\rm o}(1)}$ factor. Specifically, the fluctuations of bulk eigenvalues are bounded by $N^{-1+{\rm o}(1)}$, and the fluctuations of edge eigenvalues are bounded by $N^{-2/3+{\rm o}(1)}$. (ii) Edge universality holds for random $d$-regular graphs. That is, the distributions of $λ_2$ and $-λ_N$ converge to the Tracy-Widom$_1$ distribution associated with the Gaussian Orthogonal Ensemble. As a consequence, for sufficiently large $N$, approximately $69\%$ of $d$-regular graphs on $N$ vertices are Ramanujan, meaning $\max\{λ_2,|λ_N|\}\leq 2$.

Figures (3)

• Figure 1: An example of the local resampling performed on the graph is as follows. We replace the green edges, located on the boundary of the radius-$\ell$ neighborhood of a vertex $o$, with randomly chosen edges from the graph. Together, these edges constitute the resampling data, denoted by $\mathbf{S}$. This operation creates new blue edges and establishes a new boundary.
• Figure 2: Top Panel: In the forest $\mathcal{F}$, red edges represent core edges $\mathcal{C}$. The used core edges belong to radius-$(\ell+1)$ balls, while each unused core edge $\mathcal{C}^\circ$ forms its own connected component. Together, the red and green edges constitute the switching edges $\mathcal{K}$. Bottom Panel: We construct $\mathcal{F}^+$ from $\mathcal{F}$ by selecting an unused core edge (the rightmost red edge), expanding it into a radius-$(\ell+1)$ ball, and adding $\mu$ new switching edges.
• Figure 3: This figure illustrates the three steps involved in the proof of \ref{['p:iteration']}. First, we rewrite the product $(\widetilde{G}_{oo}^{(i)} - Y_t) \prod_{j=1}^{p-1} \widetilde{{\mathcal{W}}}_j$, which depends on the Green's functions of the switched graph ${\widetilde{{\mathcal{G}}}}$, in terms of the Green's functions of the original graph ${\mathcal{G}}$. This reformulation is expressed as $\widehat{B}_0 \prod_{j=1}^{p-1} \widehat{{\mathcal{W}}}_j$ in \ref{['l:fyibu']}. Second, we regroup the term $\prod_{j=1}^{p-1} \widehat{{\mathcal{W}}}_j$ as $(d-1)^{3h'\ell} R_{h'}$ multiplied by an admissible function, see \ref{['e:W_decompose']}. Finally, \ref{['p:iteration']} follows by discussing the three cases in the decomposition \ref{['e:first_W0']} of $\widehat{B}_0$.

Theorems & Definitions (144)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Lemma 2.1
• Lemma 2.2: MR1488333
• Lemma 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Lemma 2.7: huang2024spectrum