Exponential singularity and rank bounds for random symmetric matrices
Abstract
Let A be an n× n real symmetric random matrix whose entries above the diagonal are independent, each with largest atom at most β∈ (0,1). The diagonal entries have an arbitrary joint distribution and are independent of the entries above the diagonal. We prove that ℙ[rank(A)≤ n−k]≤ 2e−cβnk, 1≤ k≤ n, where cβ>0 depends only on β. This extends Han's corank bound to the full range 1≤ k≤ n. No moment assumptions are needed, and the entries need not be centered or identically distributed. In particular, the case k=1 gives an exponential singularity bound under the uniform atom assumption. Our argument also gives a new proof of the exponential singularity bound for symmetric Rademacher matrices obtained by Campos, Jenssen, Michelen, and Sahasrabudhe. For every fixed p∈ (0,1), it yields the exponential singularity bound conjectured by Costello and Vu for the adjacency matrix of the Erdős--Rényi random graph G(n,p), where the case p=1/2 was previously proved by Han.
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BibTeX
@misc{liang2026exponential,
title = {{Exponential singularity and rank bounds for random symmetric matrices}},
author = {Huaijin Liang and Kexin Yu and Tingzhou Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39424},
eprint = {2610.39424},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39424},
}
RIS
TY - UNPB
TI - Exponential singularity and rank bounds for random symmetric matrices
AU - Huaijin Liang
AU - Kexin Yu
AU - Tingzhou Yu
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39424
UR - https://arxiv.org/abs/2610.39424
AB - Let A be an n× n real symmetric random matrix whose entries above the diagonal are independent, each with largest atom at most β∈ (0,1). The diagonal entries have an arbitrary joint distribution and are independent of the entries above the diagonal. We prove that ℙ[rank(A)≤ n−k]≤ 2e^−c_βnk, 1≤ k≤ n, where c_β>0 depends only on β. This extends Han's corank bound to the full range 1≤ k≤ n. No moment assumptions are needed, and the entries need not be centered or identically distributed. In particular, the case k=1 gives an exponential singularity bound under the uniform atom assumption. Our argument also gives a new proof of the exponential singularity bound for symmetric Rademacher matrices obtained by Campos, Jenssen, Michelen, and Sahasrabudhe. For every fixed p∈ (0,1), it yields the exponential singularity bound conjectured by Costello and Vu for the adjacency matrix of the Erdős--Rényi random graph G(n,p), where the case p=1/2 was previously proved by Han.
ER -