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17
The littlewoodofford problem and invertibility of random matrices
 Adv. Math
"... Abstract. We prove two basic conjectures on the distribution of the smallest singular value of random n×n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n −1/2, which is optimal for Gaussian matrices. Moreover, we give a opti ..."
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Cited by 44 (10 self)
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Abstract. We prove two basic conjectures on the distribution of the smallest singular value of random n×n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n −1/2, which is optimal for Gaussian matrices. Moreover, we give a optimal estimate on the tail probability. This comes as a consequence of a new and essentially sharp estimate in the LittlewoodOfford problem: for i.i.d. random variables Xk and real numbers ak, determine the probability p that the sum � k akXk lies near some number v. For arbitrary coefficients ak of the same order of magnitude, we show that they essentially lie in an arithmetic progression of length 1/p. 1.
Random matrices: Universality of esds and the circular law
, 2009
"... Given an n × n complex matrix A, let ..."
Random matrices: The circular law
, 2008
"... Let x be a complex random variable with mean zero and bounded variance σ². Let Nn be a random matrix of order n with entries being i.i.d. 1 copies of x. Let λ1,..., λn be the eigenvalues of ..."
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Cited by 22 (9 self)
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Let x be a complex random variable with mean zero and bounded variance σ². Let Nn be a random matrix of order n with entries being i.i.d. 1 copies of x. Let λ1,..., λn be the eigenvalues of
From the LittlewoodOfford problem to the Circular Law: Universality of the spectral distribution of random matrices
 BULL. AMER. MATH. SOC
, 2009
"... The famous circular law asserts that if Mn is an n×n matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution of the normalized matrix 1 √ Mn converges both in probability and n almost surely to the uniform distribution on the unit disk {z ∈ C: z  ≤1 ..."
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Cited by 11 (1 self)
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The famous circular law asserts that if Mn is an n×n matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution of the normalized matrix 1 √ Mn converges both in probability and n almost surely to the uniform distribution on the unit disk {z ∈ C: z  ≤1}. After a long sequence of partial results that verified this law under additional assumptions on the distribution of the entries, the circular law is now known to be true for arbitrary distributions with mean zero and unit variance. In this survey we describe some of the key ingredients used in the establishment of the circular law at this level of generality, in particular recent advances in understanding the LittlewoodOfford problem and its inverse.
The Circular Law for Random Matrices
"... We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a densit ..."
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Cited by 10 (0 self)
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We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries. We assume that the entries have moment of order E Xjk  2 ϕ(x), with some positive function ϕ(x) which is growing of order (log(1 + x)) 7, or that they are sparsely nonzero. The results are based on and extend previous work of Bai, Rudelson and the authors. 1
Random matrices: A general approach for the least singular value problem, preprint
"... Abstract. Let x be a complex random variable with mean zero and bounded variance. Let Nn be the random matrix of size n whose entries are iid copies of x and M be an arbitrary matrix. We give a general estimate for the least singular value of the matrix Mn: = M + Nn. In various special cases, our es ..."
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Cited by 5 (4 self)
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Abstract. Let x be a complex random variable with mean zero and bounded variance. Let Nn be the random matrix of size n whose entries are iid copies of x and M be an arbitrary matrix. We give a general estimate for the least singular value of the matrix Mn: = M + Nn. In various special cases, our estimate extends or refines previous known results. 1.
The single ring theorem
, 2009
"... We study the empirical measure LAn of the eigenvalues of nonnormal square matrices of the form An = UnDnVn with Un,Vn independent Haar distributed on the unitary group and Dn real diagonal. We show that when the empirical measure of the eigenvalues of Dn converges, and Dn satisfies some technical c ..."
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Cited by 5 (2 self)
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We study the empirical measure LAn of the eigenvalues of nonnormal square matrices of the form An = UnDnVn with Un,Vn independent Haar distributed on the unitary group and Dn real diagonal. We show that when the empirical measure of the eigenvalues of Dn converges, and Dn satisfies some technical conditions, LAn converges towards a rotationally invariant measure on the complex plane whose support is a single ring. In particular, we provide a complete proof of FeinbergZee single ring theorem [5]. We also consider the case where Un,Vn are independent Haar distributed on the orthogonal group. 1 The problem Horn [15] asked the question of describing the eigenvalues of a square matrix with prescribed singular values. If A is a n × n matrix with singular values s1 ≥... ≥
Circular law theorem for random Markov matrices, Probab
 Theory Related Fields
"... Abstract. Let (Xjk)jk�1 be i.i.d. nonnegative random variables with bounded density, mean m, and finite positive variance σ 2. Let M be the n × n random Markov matrix with i.i.d. rows defined by Mjk = Xjk/(Xj1+···+Xjn). In particular, when X11 follows an exponential law, the random matrix M belongs ..."
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Cited by 3 (0 self)
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Abstract. Let (Xjk)jk�1 be i.i.d. nonnegative random variables with bounded density, mean m, and finite positive variance σ 2. Let M be the n × n random Markov matrix with i.i.d. rows defined by Mjk = Xjk/(Xj1+···+Xjn). In particular, when X11 follows an exponential law, the random matrix M belongs to the Dirichlet Markov Ensemble of random stochastic matrices. Let λ1,...,λn be the eigenvalues of √ nM i.e. the roots in C of its characteristic polynomial. Our main result states that with probability one, the counting probability measure 1 1 δλ1 n nδλn converges weakly as n → ∞ to the uniform law on the disk {z ∈ C: z  � m −1 σ}. The bounded density assumption is purely technical and comes from the way we control the operator norm of the resolvent.
Circular law and arc law for truncation of random unitary matrix
 Journal of Mathematical Physics
"... Let V be the m × m upperleft corner of an n × n Haarinvariant unitary matrix. Let λ1, · · · , λm be the eigenvalues of V. We prove that the empirical distribution of a normalization of λ1, · · · , λm goes to the circular law, that is, the uniform distribution on {z ∈ C; z  ≤ 1} as m → ∞ ..."
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Cited by 3 (3 self)
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Let V be the m × m upperleft corner of an n × n Haarinvariant unitary matrix. Let λ1, · · · , λm be the eigenvalues of V. We prove that the empirical distribution of a normalization of λ1, · · · , λm goes to the circular law, that is, the uniform distribution on {z ∈ C; z  ≤ 1} as m → ∞ with m/n → 0. We also prove that the empirical distribution of λ1, · · · , λm goes to the arc law, that is, the uniform distribution on {z ∈ C; z  = 1} as m/n → 1. These explain two observations by ˙ Zyczkowski and Sommers (2000).