Bernoulli flow for Erdős-Rényi graphs
Abstract
We study the eigenvalues and eigenvectors of the adjacency matrix A of the Erdős-Rényi graph 𝔾(N,p) in the regime Np ≫ (log N)2. We establish optimal isotropic delocalization for the bulk eigenvectors u, meaning that ⟨ v, u⟩2 ≤ (C log N)/N with very high probability for any deterministic normalized v. In addition, we prove local spectral universality in the bulk by showing that the local spectral statistics coincide with those of the GOE. The main tool of our proof is a local law for the resolvent of A, down to optimal spectral scales and with optimal error bounds. Its proof relies on a new approach to local laws that we call the Bernoulli flow. It is a characteristic flow method in which the usual Brownian process is replaced by a Bernoulli process, where each edge of the graph forms an independent Markov process that jumps at unit rate from closed to open. The spectral parameter flows according to a suitably constructed matrix-valued Bernoulli characteristic flow.
Related
- Delocalized region of bipartite Erdős-Rényi graphs
- The spectral edge of sparse directed Erdös-Rényi graphs
- Local Search for Almost-Spanning Square Grids in Erdős--Rényi Random Graphs
- Non-Adaptive Learning of Sparse Erdős--Rényi Graphs via Affine Splitting
- Majority Dynamics on Resampled Sparse Erdős--Rényi Graphs: Gaussian Winner Selection in Collaborative Agent Networks
BibTeX
@misc{henheik2026bernoulli,
title = {{Bernoulli flow for Erdős-Rényi graphs}},
author = {Joscha Henheik and Antti Knowles},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15566},
eprint = {2609.15566},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15566},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bernoulli flow for Erdős-Rényi graphs
AU - Joscha Henheik
AU - Antti Knowles
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15566
UR - https://arxiv.org/abs/2609.15566
AB - We study the eigenvalues and eigenvectors of the adjacency matrix A of the Erdős-Rényi graph 𝔾(N,p) in the regime Np ≫ (log N)^2. We establish optimal isotropic delocalization for the bulk eigenvectors u, meaning that ⟨ v, u⟩^2 ≤ (C log N)/N with very high probability for any deterministic normalized v. In addition, we prove local spectral universality in the bulk by showing that the local spectral statistics coincide with those of the GOE. The main tool of our proof is a local law for the resolvent of A, down to optimal spectral scales and with optimal error bounds. Its proof relies on a new approach to local laws that we call the Bernoulli flow. It is a characteristic flow method in which the usual Brownian process is replaced by a Bernoulli process, where each edge of the graph forms an independent Markov process that jumps at unit rate from closed to open. The spectral parameter flows according to a suitably constructed matrix-valued Bernoulli characteristic flow.
ER -