We study m by m random matrices M with jointly Gaussian entries. Assuming a global small-
ball probability bound inf x, y∈ S m− 1 ℙ⎛⎝⎪⎪ x* M y⎪⎪> m− O (1)⎞⎠≥ 1/2 and a …

Let ξ ξ be a non-constant real-valued random variable with finite support and let M_ n (ξ) M n
(ξ) denote an n * nn× n random matrix with entries that are independent copies of ξ ξ. For ξ ξ …

The problem of estimating the smallest singular value of random square matrices is
important in connection with matrix computations and analysis of the spectral distribution. In …

Matrix perturbation bounds (such as Weyl and Davis-Kahan) are frequently used in many
branches of mathematics. Most of the classical results in this area are optimal, in the worst …

Let $ p\in (0, 1/2) $ be fixed, and let $ B_n (p) $ be an $ n\times n $ random matrix with iid
Bernoulli random variables with mean $ p $. We show that for all $ t\ge 0$,\[\mathbb {P}[s_n …

Let R n be an n× n random matrix with iid subgaussian entries. Let M be an n× n
deterministic matrix with norm‖ M‖≤ n γ where 1/2< γ< 1. The goal of this paper is to give …

In this thesis I aim to show several developments related to notions of randomness and
structure in combinatorics and probability. One central notion, the pseudorandomness …

Let Q n, d denote the random combinatorial matrix whose rows are independent of one
another and such that each row is sampled uniformly at random from the subset of vectors in …

Non-asymptotic random matrix theory focuses on obtaining quantitative high-probability
estimates for the spectral properties of random matrices with large but fixed sizes. Such …

Random Matrices, Additive Combinatorics, and Convex Geometry Page 1 INSTITUTO DE
MATEMATICA PURA E APLICADA Random Matrices, Additive Combinatorics, and Convex …