Benjamini-Schramm vs Plancherel convergence

TL;DR

This paper compares two types of convergence for graphs or groups, demonstrating that Plancherel convergence is a strictly stronger notion than Benjamini-Schramm convergence, using criteria based on length spectrum counting functions.

Contribution

The paper introduces criteria for Plancherel and Benjamini-Schramm convergence via length spectrum counts and establishes the strict hierarchy between these convergence notions.

Findings

Plancherel convergence is strictly stronger than Benjamini-Schramm convergence.

New criteria for convergence types based on length spectrum counting functions.

Clarifies the relationship between spectral and local convergence notions.

Abstract

We show that Plancherel convergence is strictly stronger than Benjamini-Schramm convergence. In order to do so, we introduce two criteria for Plancherel and Benjamini-Schramm convergence in terms of counting functions of the length spectrum.