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Let $M_n$ be a random $ntimes n$ matrix with i.i.d. $text{Bernoulli}(1/2)$ entries. We show that for fixed $kge 1$, [lim_{nto infty}frac{1}{n}log_2mathbb{P}[text{corank }M_nge k] = -k.]
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Let $xi$ be a non-constant real-valued random variable with finite support, and let $M_{n}(xi)$ denote an $ntimes n$ random matrix with entries that are independent copies of $xi$. For $xi$ which is not uniform on its support, we show that begin{align*} mathbb{P}[M_{n}(xi)text{ is singular}] &= mathbb{P}[text{zero row or column}] + (1+o_n(1))mathbb{P}[text{two equal (up to sign) rows or columns}], end{align*} thereby confirming a folklore conjecture. As special cases, we obtain: (1) For $xi = text{Bernoulli}(p)$ with fixed $p in (0,1/2)$, [mathbb{P}[M_{n}(xi)text{ is singular}] = 2n(1-p)^{n} + (1+o_n(1))n(n-1)(p^2 + (1-p)^2)^{n},] which determines the singularity probability to two asymptotic terms. Previously, no result of such precision was available in the study of the singularity of random matrices. (2) For $xi = text{Bernoulli}(p)$ with fixed $p in (1/2,1)$, [mathbb{P}[M_{n}(xi)text{ is singular}] = (1+o_n(1))n(n-1)(p^2 + (1-p)^2)^{n}.] Previously, only the much weaker upper bound of $(sqrt{p} + o_n(1))^{n}$ was known due to the work of Bourgain-Vu-Wood. For $xi$ which is uniform on its support: (1) We show that begin{align*} mathbb{P}[M_{n}(xi)text{ is singular}] &= (1+o_n(1))^{n}mathbb{P}[text{two rows or columns are equal}]. end{align*} (2) Perhaps more importantly, we provide a sharp analysis of the contribution of the `compressible part of the unit sphere to the lower tail of the smallest singular value of $M_{n}(xi)$.
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 $C > 1$.
We show that the contact process on the rank-one inhomogeneous random graphs and Erdos-R{e}nyi graphs with mean degree large enough survives a time exponential in the size of these graphs for any positive infection rate. In addition, a metastable result for the extinction time is also proved.
We determine the rank of a random matrix over an arbitrary field with prescribed numbers of non-zero entries in each row and column. As an application we obtain a formula for the rate of low-density parity check codes. This formula vindicates a conjecture of Lelarge (2013). The proofs are based on coupling arguments and a novel random perturbation, applicable to any matrix, that diminishes the number of short linear relations.
We survey recent mathematical results about the spectrum of random band matrices. We start by exposing the Erd{H o}s-Schlein-Yau dynamic approach, its application to Wigner matrices, and extension to other mean-field models. We then introduce random band matrices and the problem of their Anderson transition. We finally describe a method to obtain delocalization and universality in some sparse regimes, highlighting the role of quantum unique ergodicity.
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