On the singularity of random symmetric matrices

Campos, Marcelo; Mattos, Letícia; Morris, Robert; Morrison, Natasha

doi:10.1215/00127094-2020-0054

Cited by 19 publications

(38 citation statements)

References 21 publications

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“…In a recent work [10], the first two named authors utilized and extended some of the ideas introduced here in order to provide the best‐known upper bound for the well‐studied problem of estimating the singularity probability of random symmetric

{\pm 1}

‐valued matrices. (After this work had been completed, Campos, Mattos, Morris, and Morrison [2] extended and refined these ideas further and obtained an even stronger bound on this singularity problem. )…”

Section: Introductionmentioning

confidence: 99%

On the counting problem in inverse Littlewood–Offord theory

Ferber

Jain

Luh

et al. 2020

J. London Math. Soc.

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Let ε1,…,εn be independent and identically distributed Rademacher random variables taking values ±1 with probability 1/2 each. Given an integer vector a=(a1,…,an), its concentration probability is the quantity ρfalse(bold-italicafalse):=trueprefixsupx∈ZPrfalse(ε1a1+⋯+εnan=xfalse). The Littlewood–Offord problem asks for bounds on ρ(a) under various hypotheses on bold-italica, whereas the inverse Littlewood–Offord problem, posed by Tao and Vu, asks for a characterization of all vectors bold-italica for which ρ(a) is large. In this paper, we study the associated counting problem: How many integer vectors bold-italica belonging to a specified set have large ρ(a)? The motivation for our study is that in typical applications, the inverse Littlewood–Offord theorems are only used to obtain such counting estimates. Using a more direct approach, we obtain significantly better bounds for this problem than those obtained using the inverse Littlewood–Offord theorems of Tao and Vu and of Nguyen and Vu. Moreover, we develop a framework for deriving upper bounds on the probability of singularity of random discrete matrices that utilizes our counting result. To illustrate the methods, we present the first ‘exponential‐type’ (that is, exp(−cnc) for some positive constant c) upper bounds on the singularity probability for the following two models: (i) adjacency matrices of dense signed random regular digraphs, for which the previous best‐known bound is O(n−1/4), due to Cook; and (ii) dense row‐regular {0,1}‐matrices, for which the previous best‐known bound is OCfalse(n−Cfalse) for any constant C>0, due to Nguyen.

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{\pm 1}

Section: Introductionmentioning

confidence: 99%

On the counting problem in inverse Littlewood–Offord theory

Ferber

Jain

Luh

et al. 2020

J. London Math. Soc.

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show abstract

“…due to Campos, Mattos, Morris, and Morrison [1]. Moreover, as was noted in [1], this bound is the best one can hope to obtain using the existing technique.…”

Section: Introductionmentioning

confidence: 55%

“…due to Campos, Mattos, Morris, and Morrison [1]. Moreover, as was noted in [1], this bound is the best one can hope to obtain using the existing technique. The common belief is that p(n) = ( 1 2 + o(1)) n , which, if true, is clearly best possible, as one can check by calculating the probability that M n has at least two identical rows/columns.…”

Section: Introductionmentioning

confidence: 55%

“…Let K := |K er (M )| be the random variable that outputs the size of the kernel of M over Z q , and we will show, using an elegant Fourier-type argument, that E[K ] ≤ 2+o(1). Then, observe that if M is singular, then K ≥ q, so by Markov's inequality we obtain that the probability to be singular is at most 2+o (1) q . In fact, the same argument gives a bound of 1+o( 1) q−1 (if we define K to be the number of non-zero vectors in the kernel), but it does not really matter for us as our bound is far from being the conjectured (or even the best known) bound.…”

Section: Introductionmentioning

confidence: 96%

“…In this model, even proving that p(n) = o(1) (this problem was posed by Weiss in the early 1990s) is quite challenging and was only settled in 2005 by Costello, Tao, and Vu [2], who showed that p(n) = O n −1/8+o (1) .…”

Section: Introductionmentioning

confidence: 99%

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