Year: 2026
Type: preprint
Abstract: We resolve the Rademacher singularity conjecture: an $n\times n$ matrix with independent uniform $\{-1,1\}$ entries is singular with probability $(2+o(1))n^22^{-n}$. More generally, for an $n\times n$ matrix $M_n$ with independent entries uniform on a fixed set $S\subset\mathbb{R}$ of cardinality $q\ge2$, we prove $\mathbb{P}(s_n(M_n)\le z/\sqrt{n}) \le Cz+a_n(S)+C'n^{1+\varepsilon}q^{-n}$ for all $n\ge1$, $z\ge0$, and $\varepsilon>0$, with $C=C(S)$ and $C'=C'(S,\varepsilon)$. 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.
Source: arXiv (Cornell University)
Authors: Hu, Xuanang, Song, Zeyan, Wu, Xinglong
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7215126740