Inverse Littlewood-Offord problems and The Singularity of Random Symmetric Matrices
Inverse Littlewood-Offord problems and The Singularity of Random Symmetric Matrices
复制标题
Littlewood-Offford 逆问题和随机对称矩阵的奇异性
DOI：
10.1215/00127094-1548344
复制
发表时间：
2011
期刊：
影响因子：
--
通讯作者：
H. Nguyen
中科院分区：
文献类型：
--
作者：
H. Nguyen
设M_n$表示一个随机对称n × n$矩阵，其上对角元为iid Bernoulli随机变量（取值为-1和1的概率为1/2）。本文改进了Costello，Tao和Vu的结果，证明了对于任意正常数$C$，$M_n$以概率$1-O（n^{-C}）$是非奇异的。证明使用二次形式的逆Littlewood-Offord结果，这是它自己的兴趣。
Let $M_n$ denote a random symmetric $n$ by $n$ matrix, whose upper diagonal entries are iid Bernoulli random variables (which take value -1 and 1 with probability 1/2). Improving the earlier result by Costello, Tao and Vu, we show that $M_n$ is non-singular with probability $1-O(n^{-C})$ for any positive constant $C$. The proof uses an inverse Littlewood-Offord result for quadratic forms, which is of interest of its own.