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Sharp singularity asymptotics for discrete random matrices
Abstract
We resolve the Rademacher singularity conjecture: an n× n matrix with independent uniform {−1,1} entries is singular with probability (2+o(1))n22−n. More generally, for an n× n matrix Mn with independent entries uniform on a fixed set S⊂ℝ of cardinality q≥2, we prove ℙ(sn(Mn)≤ z/√n) ≤ Cz+an(S)+C'n1+εq−n for all n≥1, z≥0, and ε>0, with C=C(S) and C'=C'(S,ε). Here sn denotes the least singular value, and an(S) is the sum of the probabilities of the events involving zero rows or columns and equal or opposite pairs of rows or columns.
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BibTeX
@misc{hu2026sharp,
title = {{Sharp singularity asymptotics for discrete random matrices}},
author = {Xuanang Hu and Zeyan Song and Xinglong Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.39463},
eprint = {2609.39463},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39463},
}
RIS
TY - UNPB
TI - Sharp singularity asymptotics for discrete random matrices
AU - Xuanang Hu
AU - Zeyan Song
AU - Xinglong Wu
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2609.39463
UR - https://arxiv.org/abs/2609.39463
AB - We resolve the Rademacher singularity conjecture: an n× n matrix with independent uniform {−1,1} entries is singular with probability (2+o(1))n^22^−n. More generally, for an n× n matrix M_n with independent entries uniform on a fixed set S⊂ℝ of cardinality q≥2, we prove ℙ(s_n(M_n)≤ z/√n) ≤ Cz+a_n(S)+C'n^1+εq^−n for all n≥1, z≥0, and ε>0, with C=C(S) and C'=C'(S,ε). Here s_n denotes the least singular value, and a_n(S) is the sum of the probabilities of the events involving zero rows or columns and equal or opposite pairs of rows or columns.
ER -