Journal
ANNALS OF PROBABILITY
Volume 47, Issue 5, Pages 3278-3302Publisher
INST MATHEMATICAL STATISTICS
DOI: 10.1214/19-AOP1339
Keywords
Erdos-Renyi graph; sparse random matrix; local law; eigenvector delocalization
Categories
Funding
- Swiss National Science Foundation
- European Research Council
Ask authors/readers for more resources
We prove a local law for the adjacency matrix of the Erdos-Renyi graph G(N, p) in the supercritical regime pN >= C logN where G(N, p) has with high probability no isolated vertices. In the same regime, we also prove the complete delocalization of the eigenvectors. Both results are false in the complementary subcritical regime. Our result improves the corresponding results from (Ann. Probab. 41 (2013) 2279-2375) by extending them all the way down to the critical scale pN = O(logN). A key ingredient of our proof is a new family of multilinear large deviation estimates for sparse random vectors, which carefully balance mixed l(2) and l(infinity) norms of the coefficients with combinatorial factors, allowing us to prove strong enough concentration down to the critical scale pN = O(logN). These estimates are of independent interest and we expect them to be more generally useful in the analysis of very sparse random matrices.
Authors
I am an author on this paper
Click your name to claim this paper and add it to your profile.
Reviews
Recommended
No Data Available