Singularity of random symmetric matrices – simple proof

Ferber, Asaf

doi:10.5802/crmath.215

Cited by 6 publications

(3 citation statements)

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“…This is accomplished in the key Proposition 2·5. Finally, we note that an upper bound on the singularity probability of the form can be deduced for from estimates on the expected size of the kernel over due to [ 9 ], but for similar reasons to those mentioned above these estimates do not appear to extend to p which is larger than a small polynomial.…”

Section: Introductionmentioning

confidence: 73%

Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial

Ferber¹,

Jain²,

Sah³

et al. 2022

Math. Proc. Camb. Phil. Soc.

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Conditional on the extended Riemann hypothesis, we show that with high probability, the characteristic polynomial of a random symmetric $\{\pm 1\}$ -matrix is irreducible. This addresses a question raised by Eberhard in recent work. The main innovation in our work is establishing sharp estimates regarding the rank distribution of symmetric random $\{\pm 1\}$ -matrices over $\mathbb{F}_p$ for primes $2 < p \leq \exp(O(n^{1/4}))$ . Previously, such estimates were available only for $p = o(n^{1/8})$ . At the heart of our proof is a way to combine multiple inverse Littlewood–Offord-type results to control the contribution to singularity-type events of vectors in $\mathbb{F}_p^{n}$ with anticoncentration at least $1/p + \Omega(1/p^2)$ . Previously, inverse Littlewood–Offord-type results only allowed control over vectors with anticoncentration at least $C/p$ for some large constant $C > 1$ .

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Section: Introductionmentioning

confidence: 73%

Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial

Ferber¹,

Jain²,

Sah³

et al. 2022

Math. Proc. Camb. Phil. Soc.

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“…Since z 0 = 3 2 q 2 (s + t) log q ≤ 3q 2 t log q and s = 𝜔(log t), the above quantity is o(1). We have then established (5), which combined with (4) gives the desired conclusion. ▪…”

Section: Proofs Of Key Lemmasmentioning

confidence: 97%

“…The main difference between our approach and that of Scott is that when working with modulus

q>2

, the arguments are more involved and more amenable to discrete Fourier analysis. Even though we do not believe that all of the Fourier‐type lemmas appearing in this paper are new, we included full proofs as they may be of independent interest (for example, for an application of these lemmas to the singularity problem of random symmetric Bernoulli matrices, the reader is referred to Reference [5]).…”

Section: Introductionmentioning

confidence: 99%

On subgraphs with degrees of prescribed residues in the random graph

Ferber

Hardiman

Krivelevich

2022

Random Struct Algorithms

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We show that with high probability the random graph Gn,1false/2$$ {G}_{n,1/2} $$ has an induced subgraph of linear size, all of whose degrees are congruent to r0.3emfalse(mod0.3emqfalse)$$ r\kern0.3em \left(\operatorname{mod}\kern0.3em q\right) $$ for any fixed r$$ r $$ and q≥2$$ q\ge 2 $$. More generally, the same is true for any fixed distribution of degrees modulo q$$ q $$. Finally, we show that with high probability we can partition the vertices of Gn,1false/2$$ {G}_{n,1/2} $$ into q+1$$ q+1 $$ parts of nearly equal size, each of which induces a subgraph all of whose degrees are congruent to r0.3emfalse(mod0.3emqfalse)$$ r\kern0.3em \left(\operatorname{mod}\kern0.3em q\right) $$. Our results resolve affirmatively a conjecture of Scott, who addressed the case q=2$$ q=2 $$.

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