# Local law and Tracy--Widom limit for sparse stochastic block models

## TL;DR

This paper establishes the local spectral law and Tracy--Widom fluctuations for extremal eigenvalues in sparse stochastic block models, revealing detailed spectral behavior in these networks.

## Contribution

It proves a local semicircle law and Tracy--Widom fluctuations for sparse stochastic block models, with explicit spectral edge formulas and outlier gap analysis.

## Key findings

- Spectral edge shift explicitly characterized.
- Extremal eigenvalues follow Tracy--Widom law.
- Large gap between outliers and spectral edge proven.

## Abstract

We consider the spectral properties of sparse stochastic block models, where $N$ vertices are partitioned into $K$ balanced communities. Under an assumption that the intra-community probability and inter-community probability are of similar order, we prove a local semicircle law up to the spectral edges, with an explicit formula on the deterministic shift of the spectral edge. We also prove that the fluctuation of the extremal eigenvalues is given by the GOE Tracy--Widom law after rescaling and centering the entries of sparse stochastic block models. Applying the result to sparse stochastic block models, we rigorously prove that there is a large gap between the outliers and the spectral edge without centering.