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Rank and Invertibility of Dense Signed Random Regular Matrices
Abstract
Let A be the (non-symmetric) adjacency matrix of a uniformly random d-regular directed graph on n vertices, and let Ξ be independent of A with i.i.d. Rademacher entries. Suppose that min(d,n−d)≥λn for some fixed λ∈(0,1/2]. We show that there exists c > 0, depending only on λ, such that PA,Ξ{rank(A∘Ξ)≤ n−k}≤ e−c nk, 1≤ k≤ n. As an ingredient in the proof of the rank bound, we use the case κ=0 of the following quantitative smallest singular value estimate: PA,Ξ{sn(A∘Ξ)≤κ} ≤ Cκ√n+e−c' n, κ≥0, where C,c'>0 depend only on λ.
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BibTeX
@misc{liang2026rank,
title = {{Rank and Invertibility of Dense Signed Random Regular Matrices}},
author = {Huaijin Liang and Yuxiao Shi and Kexin Yu and Tingzhou Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05938},
eprint = {2610.05938},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05938},
}
RIS
TY - UNPB
TI - Rank and Invertibility of Dense Signed Random Regular Matrices
AU - Huaijin Liang
AU - Yuxiao Shi
AU - Kexin Yu
AU - Tingzhou Yu
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05938
UR - https://arxiv.org/abs/2610.05938
AB - Let A be the (non-symmetric) adjacency matrix of a uniformly random d-regular directed graph on n vertices, and let Ξ be independent of A with i.i.d. Rademacher entries. Suppose that min(d,n−d)≥λn for some fixed λ∈(0,1/2]. We show that there exists c > 0, depending only on λ, such that P_A,Ξ{rank(A∘Ξ)≤ n−k}≤ e^−c nk, 1≤ k≤ n. As an ingredient in the proof of the rank bound, we use the case κ=0 of the following quantitative smallest singular value estimate: P_A,Ξ{s_n(A∘Ξ)≤κ} ≤ Cκ√n+e^−c' n, κ≥0, where C,c'>0 depend only on λ.
ER -