Mathematics#19 in June 2026 EditionExtreme value statistics for eigenvalues of sparse random symmetric matrices
Anand Sundar, Helene Dupont, Viktor Kozlov et al.
OPENSTEM EDITORIAL SUMMARY
What the research examines
This paper proves that the largest eigenvalue of an Erdős-Rényi sparse random graph with average degree p = d/N converges to a Tracy-Widom distribution down to the critical sparsity limit d = Omega(log N).
Why it may matter
Establishes universal edge spectral limits critical for network community detection, random matrix theory, and quantum graph models.
What to examine carefully
When sparsity falls below log N, localized eigenvector states emerge, violating Tracy-Widom universality.
Editorial scope: This is a concise, independently written introduction based on the publisher information, not a full systematic review or independent replication. It must not replace reading the original methods, limitations and conclusions.
STUDENT RESEARCH LENS · ORIGINAL OPENSTEM NOTES
Try a small, reproducible follow-up
Simulate top eigenvalues of 2,000x2,000 sparse symmetric random matrices and fit empirical distributions to the Tracy-Widom law.
Questions to investigate
- How did the authors control non-backtracking walk cycles on localized sparse subgraphs?
- What Green's function resolvent bounds were established near the spectral edge?
These are independently proposed educational investigations, not activities that the original research authors claim to have completed.
Publication & researcher information
JournalAnnals of Probability
Published online2026-06-20
Authors (selected)Anand Sundar, Helene Dupont, Viktor Kozlov et al.
University / laboratoryNew York University, Courant Institute
CountryUnited States
Multi-author research may involve universities in several countries. Unverified institutions and contact addresses are not inferred from authors' names.