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Let $A$ be an $ntimes n$ random matrix whose entries are i.i.d. with mean $0$ and variance $1$. We present a deterministic polynomial time algorithm which, with probability at least $1-2exp(-Omega(epsilon n))$ in the choice of $A$, finds an $epsilon n times epsilon n$ sub-matrix such that zeroing it out results in $widetilde{A}$ with [|widetilde{A}| = Oleft(sqrt{n/epsilon}right).] Our result is optimal up to a constant factor and improves previous results of Rebrova and Vershynin, and Rebrova. We also prove an analogous result for $A$ a symmetric $ntimes n$ random matrix whose upper-diagonal entries are i.i.d. with mean $0$ and variance $1$.
The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergen
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
We give an asymptotic evaluation of the complexity of spherical p-spin spin-glass models via random matrix theory. This study enables us to obtain detailed information about the bottom of the energy landscape, including the absolute minimum (the grou
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.]
This is a brief survey of classical and recent results about the typical behavior of eigenvalues of large random matrices, written for mathematicians and others who study and use matrices but may not be accustomed to thinking about randomness.