Inverse Littlewood-Offord problems and The Singularity of Random Symmetric Matrices

By Hoi H. Nguyen. 2011

64 citations recorded by the source corpus.

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.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/85aaadf25382569249d32063d446dd95f527f48c

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In Mathematical Physics

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Citations in the corpus

Small Ball Probability, Inverse Theorems, and Applications (2012, methodology): “The first superpolynomial bound was obtained by the first author [36] very recently.”
The Elliptic Law (2012, background): “It seems plausible that after an appropriate linear transform we can trap most of the entries aij of An into a GAP of small size and small rank (in the spirit of [32]).”
Adjacency matrices of random digraphs: singularity and anti-concentration (2015, background): “The corresponding problem for symmetric Bernoulli matrices was considered in [11], [27], [36].”