Lecture Notes on Edge Universality for Random Regular Graphs

TL;DR

The work studies edge universality for random $d$-regular graphs by combining a local law for Green's functions with a three-step strategy: derive self-consistent equations, refine them microscopically via loop equations, and implement a Dyson Brownian-motion–style comparison to GOE at the edge. Central tools include local resampling to construct exchangeable pairs, and meticulous resolvent expansions (Schur complement and Woodbury) organized by forests and admissible functions. The authors prove that the self-consistent relations hold with optimal error and establish a microscopic loop equation that drives the Green’s-function comparison near the spectral edge. This framework yields edge universality: the second-largest eigenvalue fluctuations converge to Tracy–Widom$_1$, with GOE edge statistics governing the entire edge spectrum, and implies Ramanujan-type spectral gaps with high probability. The approach integrates precise combinatorial forest constructions, resolvent calculus, and probabilistic concentration to bridge sparse random graphs with universal random-matrix behavior.

Abstract

The purpose of this note is to explain the structure, general strategy, and main ideas of the proof in the work of Huang, McKenzie, and Yau (2024) on the Ramanujan property and edge universality of random regular graphs. The core of the argument is the derivation of self-consistent equations and a microscopic version of the loop equations for random $d$-regular graphs. We first recall the local law for random $d$-regular graphs, and then illustrate the main ideas behind the derivation of the self-consistent equations and the first loop equation.

Figures (7)

• Figure 1: Left panel: for a truncated $d$-regular tree ${\mathcal{T}}$, we attach a weight $-\Delta$ to each leaf vertex; middle panel: for a general graph ${\mathcal{T}}$, we attach a weight $-(d-D_{ii}({\mathcal{T}}))/(d-1)$ to each vertex; right panel: remove the vertex set ${\mathbb X}=\{i\}$.
• Figure 2: Extreme eigenvalues of Gaussian $\beta$-ensemble converge to the Airy$_\beta$ point process.
• Figure 3: An example of the local resampling performed on the graph is as follows. We replace the blue 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 red edges and establishes a new boundary.
• Figure 4: We view ${\mathcal{F}}$ (corresponding to the solid edges) as an embedded subgraph of ${\mathcal{G}}$, which contains all the switching edges. The indicator $I({\mathcal{F}},{\mathcal{G}})$ signifies that $o, c_1, c_2, \ldots, c_\mu$ have tree neighborhoods and are well separated.
• Figure 5: The index set ${\mathsf A}_i := \{ \alpha \in [\![{\mu}]\!] : \mathop{\mathrm{dist}}\nolimits_{{\mathcal{T}}}(i, l_\alpha) = \ell+1 \}$ is given by $\{1,2,\cdots, 2\nu\}$, corresponding to these red nodes.

Theorems & Definitions (78)

• Theorem 1.1: Edge eigenvalue universality huang2024ramanujan
• Corollary 1.2: huang2024ramanujan
• Definition 3.1
• Proposition 3.2: huang2024spectrum
• Proposition 3.3: huang2024spectrum
• Definition 3.4
• Proposition 3.5
• proof : Proof of Proposition \ref{['p:recurbound']}
• Theorem 3.6: huang2024spectrum
• Definition 3.7