Sparse graphs and their Benjamini-Schramm limits: a spectral tour

TL;DR

This paper surveys the spectral implications of Benjamini–Schramm convergence for sparse marked graphs, with a focus on Schreier and covering graphs. It develops a framework of local operators via the group-algebra and proves BS-continuity of the average spectral measure, explores the spectral decomposition on unimodular limits, and analyzes edge eigenvalues using Alon–Boppana-type results. A key highlight is the establishment and exploitation of strong convergence in distribution for random representations of free groups, which rules out spectral outliers and yields spectrum-convergence results to universal covers for Schreier and covering graphs. Together, these results illuminate how local geometric data dictates global spectral properties in random and deterministic sparse graphs, with implications for Ramanujan-type phenomena, graph distance behavior, and broader operator-algebra connections.

Abstract

Sparse graphs with bounded average degree form a rich class of discrete structures where local geometry strongly influences global behavior. The Benjamini-Schramm (BS) convergence offers a natural framework to describe their asymptotic local structure. In this note, we survey spectral aspects of BS convergence and their applications, with a focus on random Schreier graphs and covering graphs. We review some recent progress on the spectral decomposition of the local operators on graphs. We discuss the behavior of extreme eigenvalues and the growing role of strong convergence in distribution, which rules out spectral outliers. We also give a new application of strong convergence to the typical graph distance between vertices in Schreier graphs

Figures (6)

• Figure 2.1: On the left, a graph $G$ with two connected components, on the right, the associated unlabeled rooted graphs. $U(G)$ is the uniform measure on the three displayed unlabeled rooted graphs.
• Figure 2.2: Left: $3$-regular infinite tree. Right: A realization of a Galton-Watson tree with $\mathrm {Poi}(2)$ distribution.
• Figure 2.3: A graph, a $2$-lift and their universal covering tree.
• Figure 2.4: A base graph on the left and a $3$-cover on the right. Each vertex of the base graph has a discrete fiber of $3$ elements. The permutation of each edge encodes how elements are connected.
• Figure 3.1: Left: histogram of the eigenvalues of the adjacency operator of a uniformly random $4$-regular graph with $n= 4000$ vertices, in red the density of its limit \ref{['eq:kesten']}. Right: histogram of the eigenvalues of the adjacency operator of an Erdős-Rényi graph with $n = 4000$ vertices and edge probability $4/n$.

Theorems & Definitions (28)

• Example 2.1: Conductance graphs
• Example 2.2: Cayley graphs
• Example 2.3: Schreier graphs
• Definition 2.4
• Example 2.5: Box in $\mathbb {Z}^d$
• Example 2.6: Regular graphs with large girth
• Example 2.7: Erdős-Rényi random graph and percolation in growing dimension
• Lemma 2.8
• Example 2.9: Congruence subgroups
• Example 2.10: Random actions of the free group