Lecture Notes on Edge Universality for Random Regular Graphs

TL;DR

This paper explains the proof strategy and main ideas behind establishing edge universality and Ramanujan property for random regular graphs, focusing on self-consistent equations and loop equations.

Contribution

It provides a detailed exposition of the structure and key techniques used in proving edge universality for random regular graphs, including local laws and loop equations.

Findings

Derivation of self-consistent equations for random regular graphs

Establishment of local laws for spectral distribution

Explanation of loop equations and their role in universality

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.