## From the Littlewood-Offord problem to the Circular Law: Universality of the spectral distribution of random matrices (2009)

Venue: | BULL. AMER. MATH. SOC |

Citations: | 13 - 2 self |

### BibTeX

@ARTICLE{Tao09fromthe,

author = {Terence Tao and Van Vu},

title = {From the Littlewood-Offord problem to the Circular Law: Universality of the spectral distribution of random matrices},

journal = {BULL. AMER. MATH. SOC},

year = {2009},

pages = {377--396}

}

### OpenURL

### Abstract

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 Littlewood-Offord problem and its inverse.

### Citations

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Numerical Linear Algebra
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Citation Context ...nvertible, we set κ(M) = ∞.) The condition number plays a crucial role in numerical linear algebra. The accuracy and stability of most algorithms used to solve the equation Mx = b depend on κ(M) (see =-=[5, 23]-=-, for example). The exact solution x = M −1b, in theory, can be computed quickly (by Gaussian elimination, say). However, in practice computers can only represent a finite subset of real numbers and t... |

211 | Methodologies in spectral analysis of large dimensional random matrices: a review, Statistica Sinica 9
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Citation Context ...TERENCE TAO AND VAN VU A fundamental problem in the theory of random matrices is to compute the limiting distribution of the ESD µAn of a sequence of random matrices An with sizes tending to infinity =-=[32, 4]-=-. In what follows, we consider normalized random matrices of the form An = 1 √ n Mn, where Mn = (xij)1≤i,j≤n has entries that are iid random variables xij ≡ x. The starting point of the whole area is ... |

152 | Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time
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Citation Context ...orithms proven to have polynomial complexity. There have been various attempts to explain this phenomenon. In this section we will discuss an influential recent explanation given by Spielman and Teng =-=[41, 42]-=-. 4.2. The effect of noise. An important issue in the theory of computing is noise, as almost all computational processes are effected by it. By the word noise, we would like to refer to all kinds of ... |

143 |
Random Matrices and the Statistical Theory of Energy Levels
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Citation Context ...TERENCE TAO AND VAN VU A fundamental problem in the theory of random matrices is to compute the limiting distribution of the ESD µAn of a sequence of random matrices An with sizes tending to infinity =-=[32, 4]-=-. In what follows, we consider normalized random matrices of the form An = 1 √ n Mn, where Mn = (xij)1≤i,j≤n has entries that are iid random variables xij ≡ x. The starting point of the whole area is ... |

107 | Additive Combinatorics - Tao, Vu - 2006 |

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Citation Context .... On the other hand, the right-hand side can be bounded efficiently, thanks to the fact that all di are large with overwhelming probability, which, in turn, is a consequence of Talagrand’s inequality =-=[43]-=-: Lemma 5.8 (Distance Lemma). [49, 52] With probability 1 − n−ω(1) , the distance from√ a random row vector √ to a subspace of co-dimension 1 k is at most k/n, as long as k ≫ log n. 100 Thus, with ove... |

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On a lemma of Littlewood and
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Citation Context ...AND VAN VU In their study of random polynomials, Littlewood and Offord [31] raised the question of bounding pv. They showed that if the vi are non-zero, log n then pv = O( √ ). Very soon after, Erdős =-=[13]-=-, using Sperner’s lemma, n gave a beautiful combinatorial proof for the following refinement. Theorem 3.2. Let v1, . . . , vn be non-zero numbers and ξi be i.i.d Bernoulli random variables. Then4 ( ) ... |

63 | Smallest singular value of random matrices and geometry of random polytopes. Advances in mathematics
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Citation Context ... of Mn to be iid 11 , although independence is crucially exploited in the proofs. Also, one can allow many of the entries to be 0 [50]. Remark 4.8. Results of this type first appear in [37] (see also =-=[33]-=- for some earlier related work for the least singualar value of rectangular matrices). In the special case where A = 0 and where the entries of Mn are iid and have finite fourth moment, Rudelson and V... |

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Statistical ensembles of complex, quaternion and real matrices
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Citation Context ...sy case of Conjecture 1.3 is when the entries xij of Mn are iid complex Gaussian. In this case there is the following precise formula for the joint density function of the eigenvalues, due to Ginibre =-=[17]-=- (see also [34], [25] for more discussion of this formula): ∏ (2) p(λ1, ··· ,λn) =cn |λi − λj| 2 n∏ i<j i=1 2 −n|λi| e . From here one can verify the conjecture in this case by a direct calculation. T... |

59 | Weyl groups, the hard Lefschetz theorem, and the Sperner property
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- 1980
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Citation Context ...nfirmed by Sárközy and Szemerédi [39]. Again, the bound is sharp (up to a constant factor), as can be seen by taking v1, . . . , vn to be a proper arithmetic progression such as 1, . . . , n. Stanley =-=[38]-=- gave a different proof that also classified the extremal cases. A general picture was given by Halász, who showed, among other things, that if one forbids more and more additive structure 5 in the vi... |

57 |
On the number of real roots of a random algebraic equation
- Littlewood, Offord
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(Show Context)
Citation Context ... . , ξn be i.i.d random Bernoulli variables. Define S := ∑n i=1 ξivi and pv(a) := P(S = a) and pv := supa∈Z pv(a).8 TERENCE TAO AND VAN VU In their study of random polynomials, Littlewood and Offord =-=[31]-=- raised the question of bounding pv. They showed that if the vi are non-zero, log n then pv = O( √ ). Very soon after, Erdős [13], using Sperner’s lemma, n gave a beautiful combinatorial proof for the... |

50 |
On the probability that a random 1-matrix is singular
- Kahn, Komlós, et al.
- 1995
(Show Context)
Citation Context ...Rk). Remark 3.4. Several variants of Theorem 3.2 can be found in [26, 29, 16, 27] and the references therein. The connection between the LittlewoodOfford problem and random matrices was first made in =-=[25]-=-, in connection with the question of determining how likely a random Bernoulli matrix was to be singular. The paper [25] in fact inspired much of the work of the authors described in this survey. 3.5.... |

44 |
On the asymptotic distribution of eigenvalues of random matrices
- Arnold
- 1967
(Show Context)
Citation Context ... other distributions, such as the Bernoulli distribution (in which each xij equals +1 with probability 1/2 and −1 with probability 1/2). His work has been extended and strengthened in several aspects =-=[1, 2, 34]-=-. The most general form was proved by Pastur [34]: 1 We say that a collection µn of probability measures converges to a limit µ if one has ∫ f dµn → ∫ f dµ for every continuous compactly supported fun... |

43 | On the empirical distribution of eigenvalues of large dimensional information-plus-noise type matrices
- Dozier, Silverstein
- 2007
(Show Context)
Citation Context ...t of the pole. Using techniques from probability, such as the moment method, one can show that the distributions of the singular values of 1 √n An − zI and 1 √ n Bn − zI are asymptotically the same14 =-=[3, 51, 10, 52, 11]-=-. This, however, is not sufficient to conclude that 1 log | det( √1 An − zI)| and n n 1 log | det( √1 Bn − zI)| are close. As n n remarked earlier, the main difficulty here is that some of the singula... |

43 |
The spectrum of random matrices
- Pastur
- 1972
(Show Context)
Citation Context ... other distributions, such as the Bernoulli distribution (in which each xij equals +1 with probability 1/2 and −1 with probability 1/2). His work has been extended and strengthened in several aspects =-=[1, 2, 34]-=-. The most general form was proved by Pastur [34]: 1 We say that a collection µn of probability measures converges to a limit µ if one has ∫ f dµn → ∫ f dµ for every continuous compactly supported fun... |

41 |
Smoothed analysis of algorithms
- Spielman, Teng
- 2002
(Show Context)
Citation Context ...orithms proven to have polynomial complexity. There have been various attempts to explain this phenomenon. In this section we will discuss an influential recent explanation given by Spielman and Teng =-=[41, 42]-=-. 4.2. The effect of noise. An important issue in the theory of computing is noise, as almost all computational processes are effected by it. By the word noise, we would like to refer to all kinds of ... |

39 |
The probability that a random real gaussian matrix has k real eigenvalues, and the Circular
- Edelman
- 1997
(Show Context)
Citation Context ...detailed discussion). The conjecture has been intensively worked on for many decades. The circular law was verified for the complex gaussian distribution in [32] and the real gaussian distribution in =-=[12]-=-. An approach to attack the general case was introduced in [18], leading to a resolution of the strong circular law for continuous distributions with bounded sixth moment in [3]. The sixth moment hypo... |

39 |
On the determinant of (0, 1) matrices
- Komlós
- 1967
(Show Context)
Citation Context ... 1 rows, and so should have a probability at most pv(β) of lying within β of the span of the those rows. There are some minor technical issues in making this argument (which essentially dates back to =-=[28]-=-) rigorous, arising from the fact that the n − 1 rows may be too degenerate to accurately control v, but these difficulties can be dealt with, especially if one is willing to lose factors of nO(1) in ... |

33 |
Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechtnung
- Lindeberg
- 1922
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Citation Context ...istributions do not exist. 3 Some related ideas also appear in [19]. In the context of the central limit theorem, the idea of replacing arbitrary iid ensembles by Gaussian ones goes back to Lindeberg =-=[30]-=-, and is sometimes known as the Lindeberg invariance principle; see [11] for further discussion, and a formulation of this principle for Hermitian random matrices.LITTLEWOOD-OFFORD, CIRCULAR LAW, UNI... |

32 |
Estimates for the concentration function of combinatorial number theory and probability
- Halasz
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Citation Context ... was given by Halász, who showed, among other things, that if one forbids more and more additive structure 5 in the vi, then one gets better and better bounds on pv. One corollary of his results (see =-=[24]-=- or [45, Chapter 9] is the following. Theorem 3.3. Consider v = {v1, . . . , vn}. Let Rk be the number of solutions to the equation ε1vi1 + · · · + ε2kvi2k = 0 where εi ∈ {−1, 1} and i1, . . . , i2k ∈... |

32 | On the singularity probability of random Bernoulli matrices - Tao, Vu |

31 | Inverse Littlewood-Offord theorems and the condition number of random discrete matrices
- Tao, Vu
(Show Context)
Citation Context ...se theorems from additive combinatorics, in particular Freiman’s theorem (see [15], [45, Chapter 5]) and a variant for random sums in [50, Theorem 5.2] (inspired by earlier work in [25]), the authors =-=[46]-=- brought a different view to the problem. Instead of trying to improve the bound further by imposing new assumptions, we aim to provide the full picture by finding the underlying reason for the probab... |

27 |
Foundations of a structural theory of set addition. Translated from the Russian
- Freiman
- 1973
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Citation Context ...d much of the work of the authors described in this survey. 3.5. The inverse Littlewood-Offord problem. Motivated by inverse theorems from additive combinatorics, in particular Freiman’s theorem (see =-=[15]-=-, [45, Chapter 5]) and a variant for random sums in [50, Theorem 5.2] (inspired by earlier work in [25]), the authors [46] brought a different view to the problem. Instead of trying to improve the bou... |

27 |
On a lemma of Littlewood and Offord on the distribution of certain sums
- Kleitman
- 1965
(Show Context)
Citation Context ...he sums 2 n sums S are all distinct, and so pv = 1/2 n in this case.LITTLEWOOD-OFFORD, CIRCULAR LAW, UNIVERSALITY 9 pv = Ok(n −2k−1/2 Rk). Remark 3.4. Several variants of Theorem 3.2 can be found in =-=[26, 29, 16, 27]-=- and the references therein. The connection between the LittlewoodOfford problem and random matrices was first made in [25], in connection with the question of determining how likely a random Bernoull... |

27 | Invertibility of random matrices: norm of the inverse
- Rudelson
(Show Context)
Citation Context ...ire the entries of Mn to be iid 11 , although independence is crucially exploited in the proofs. Also, one can allow many of the entries to be 0 [50]. Remark 4.8. Results of this type first appear in =-=[37]-=- (see also [33] for some earlier related work for the least singualar value of rectangular matrices). In the special case where A = 0 and where the entries of Mn are iid and have finite fourth moment,... |

20 | Circular law, Extreme Singular values and Potential theory
- Pan, Zhou
(Show Context)
Citation Context ...red some new ideas. In [21] the weak circular law for (possibly discrete) distributions with subgaussian moment was established, with the subgaussian condition relaxed to a fourth moment condition in =-=[33]-=- (see also [19] for an earlier result of similar nature), and then to (2 + η) th moment in [22]. Shortly before this last result, the strong circular law assuming (2+η) th moment was established in [5... |

19 | Universality for mathematical and physical systems
- Deift
- 2007
(Show Context)
Citation Context ...mation about a large random system (such as limiting distributions) does not depend on the particular distribution of the particles. This is often referred to as the universality phenomenon (see e.g. =-=[9]-=-). The most famous example of this phenomenon is perhaps the central limit theorem. In view of the universality phenomenon, one can see that Conjecture 1.3 generalizes Theorem 2.1 in the same way that... |

18 | On random±1 matrices: singularity and determinant - Tao, Vu - 2006 |

18 | On the condition number of a randomly perturbed matrix
- Tao, Vu
(Show Context)
Citation Context ...-OFFORD, CIRCULAR LAW, UNIVERSALITY 15 This problem was suggested to the authors by Spielman few years ago. Using the Weak Inverse Theorem, we were able to proved the following variant of Theorem 4.6 =-=[47]-=-. Theorem 4.7. For any constants a, c > 0, there is a constant b = b(a, c) > 0 such that the following holds. Let A be a n by n matrix such that ‖A‖ ≤ n a and let Mn be a random matrix with iid Bernou... |

16 |
On a conjecture of Erdős and a stronger form of Sperner’s theorem
- Katona
- 1966
(Show Context)
Citation Context ...he number of solutions to the equation ε1vi1 + ···+ ε2kvi2k =0, where εi ∈{−1, 1} and i1,...,i2k ∈{1, 2,...,n}. Then pv = Ok(n −2k−1/2 Rk). Remark 3.4. Several variants of Theorem 3.2 can be found in =-=[27, 30, 16, 28]-=- and the references therein. The connection between the Littlewood-Offord problem and random matrices was first made in [26], in connection with the question of determining how likely a random Bernoul... |

15 | The least singular value of a random square matrix is O(n −1/2 ). Comptes rendus-Mathématique
- Rudelson, Vershynin
- 2008
(Show Context)
Citation Context ... Vershynin 11 In practice, one would expect the noise at a large entry to have larger variance than one at a small entry, due to multiplicative effects.16 TERENCE TAO AND VAN VU [38] (see also [39], =-=[40]-=-) obtained sharp bounds for ‖(A + Mn) −1 ‖, using a somewhat different method, which relies on an inverse theorem of a slightly different nature; see Remark 3.12. The main idea behind the proof of The... |

12 | Random symmetric matrices are almost surely non-singular
- Costello, Tao, et al.
(Show Context)
Citation Context ...TTLEWOOD-OFFORD, CIRCULAR LAW, UNIVERSALITY 21 Combinatorics. Our studies of Littewood-Offord problem focus on the linear form S := ∑n i=1 vixii. What can one say about higher degree polynomials ? In =-=[6]-=-, it was shown that for a quadratic form Q := ∑ 1≤i,j≤n cijξiξj with non-zero coefficients, P(Q = z) is O(n−1/8 ). It is simple to improve this bound to O(n−1/4 ) [7]. On the other hand, we conjecture... |

12 | The circular law for random matrices
- Götze, Tikhomirov
- 2010
(Show Context)
Citation Context ...bgaussian moment was established, with the subgaussian condition relaxed to a fourth moment condition in [33] (see also [19] for an earlier result of similar nature), and then to (2 + η) th moment in =-=[22]-=-. Shortly before this last result, the strong circular law assuming (2+η) th moment was established in [51]. Finally, in a recent paper [52], the authors proved this conjecture (in both strong and wea... |

11 |
The central limit theorem for random determinants,” Theory of Probability and
- Girko
- 1979
(Show Context)
Citation Context ...ar. Another question concerns the determinant of random matrices. It is known, and not hard to prove, that log | det Mn| satisfies a central limit theorem when the entries of Mn are iid Gaussian; see =-=[20, 8]-=-. Girko [20] claimed that the same result holds for much more general models of matrices. We, however, are unable to verify his arguments. It would be nice to have an alternative proof. Acknowledgment... |

11 |
A brief survey on the spectral radius and the spectral distributionof large dimensional random matrices with iid entries. Random matrices and their applications
- Hwang
- 1986
(Show Context)
Citation Context ... 1.3 is when the entries xij of Mn are iid complex gaussian. In this case there is the following precise formula for the joint density function of the eigenvalues, due to Ginibre [17] (see also [34], =-=[25]-=- for more discussion of this formula): ∏ p(λ1, · · · , λn) = cn |λi − λj| 2 [i<j n∏ i=1 −n|λi| 2 e . (2) From here one can verify the conjecture in this case by a direct calculation. This was first do... |

11 |
Uber ein Problem von Erd}os und
- Sarkozy, Szemeredi
- 1965
(Show Context)
Citation Context ...nstance, Erdős and Moser [14] showed that if the vi are distinct, then pv = O(n −3/2 ln n). They conjectured that the logarithmic term is not necessary and this was confirmed by Sárközy and Szemerédi =-=[39]-=-. Again, the bound is sharp (up to a constant factor), as can be seen by taking v1, . . . , vn to be a proper arithmetic progression such as 1, . . . , n. Stanley [38] gave a different proof that also... |

9 |
Solution of the Littlewood-Offord problem in high dimensions
- Frankl, Füredi
- 1988
(Show Context)
Citation Context ...he sums 2 n sums S are all distinct, and so pv = 1/2 n in this case.LITTLEWOOD-OFFORD, CIRCULAR LAW, UNIVERSALITY 9 pv = Ok(n −2k−1/2 Rk). Remark 3.4. Several variants of Theorem 3.2 can be found in =-=[26, 29, 16, 27]-=- and the references therein. The connection between the LittlewoodOfford problem and random matrices was first made in [25], in connection with the question of determining how likely a random Bernoull... |

8 |
The strong circular law. Twenty years later
- Girko
(Show Context)
Citation Context ...eas. In [21] the weak circular law for (possibly discrete) distributions with subgaussian moment was established, with the subgaussian condition relaxed to a fourth moment condition in [33] (see also =-=[19]-=- for an earlier result of similar nature), and then to (2 + η) th moment in [22]. Shortly before this last result, the strong circular law assuming (2+η) th moment was established in [51]. Finally, in... |

7 | On the circular law
- Götze, Tikhomirov
(Show Context)
Citation Context ...ded sixth moment in [3]. The sixth moment hypothesis in [3] was lowered to (2 + η) th moment for any η > 0 in [4]. The removal of the hypothesis of continuous distribution required some new ideas. In =-=[21]-=- the weak circular law for (possibly discrete) distributions with subgaussian moment was established, with the subgaussian condition relaxed to a fourth moment condition in [33] (see also [19] for an ... |

7 |
The Littlewood-Offord problem and the condition number of random matrices
- Rudelson, Vershynin
- 2008
(Show Context)
Citation Context ...v ′ ∈ Ω such that all coefficients of v − v ′ do not exceed β in magnitude.12 TERENCE TAO AND VAN VU |Ω| ≤ p −n n −n/2+o(n) . (8) Remark 3.12. A related result (with different parameters) appears in =-=[36]-=-; in our notation, the probability p is allowed to be much smaller, but the net is coarser (essentially, a β √ n-net rather than a β-net). In terms of random matrices, the results in [36] are better s... |

5 | VU: Concentration of random determinants and permanent estimators
- COSTELLO, H
(Show Context)
Citation Context ...ar. Another question concerns the determinant of random matrices. It is known, and not hard to prove, that log | det Mn| satisfies a central limit theorem when the entries of Mn are iid Gaussian; see =-=[20, 8]-=-. Girko [20] claimed that the same result holds for much more general models of matrices. We, however, are unable to verify his arguments. It would be nice to have an alternative proof. Acknowledgment... |

5 | Random matrices: A general approach for the least singular value problem, preprint
- Tao, Vu
(Show Context)
Citation Context ... let Mn be a random matrix with iid Bernoulli entries. Then P(‖(A + Mn) −1 ‖ ≥ n b ) ≤ n −c . Using the stronger β-net Theorem, one can have a nearly optimal relation between the constants a, b and c =-=[48]-=-. These results extend, with the same proof, to a large variety of distributions. For example, one does not need require the entries of Mn to be iid 8 , although independence is crucially exploited in... |

4 |
On the tightest packing of sums of vectors
- Griggs, Lagarias, et al.
- 1983
(Show Context)
Citation Context ...he sums 2 n sums S are all distinct, and so pv = 1/2 n in this case.LITTLEWOOD-OFFORD, CIRCULAR LAW, UNIVERSALITY 9 pv = Ok(n −2k−1/2 Rk). Remark 3.4. Several variants of Theorem 3.2 can be found in =-=[26, 29, 16, 27]-=- and the references therein. The connection between the LittlewoodOfford problem and random matrices was first made in [25], in connection with the question of determining how likely a random Bernoull... |

4 |
The smallest singular value of a rectangular random matrix, preprint
- Rudelson, Vershynin
(Show Context)
Citation Context .... 6. Open problems Our investigation leads to open problems in several areas: 16A possible alternate approach would be to bound the intermediate singular values directly, by adapting the results from =-=[37]-=-. This would however require some additional effort; for instance, the results in [37] assume zero mean and bounded operator norm, which is not true in general when considering 1 √ An − zI for nonn ze... |

3 |
A simple invariance principle. [arXiv:math/0508213
- Chatterjee
(Show Context)
Citation Context ... - complex numbers z for which the resolvent (An − zI) −1 = ( 1 √ Mn − zI) n −1 exists but is extremely large. It is therefore of importance to obtain bounds on this resolvent, which leads one to see =-=[11]-=- for further discussion, and a formulation of this principle for Hermitian random matrices.8 TERENCE TAO AND VAN VU understand for which vectors v ∈ C n is (An − zI)v likely to be small. Expanding ou... |

2 |
Elementary problems and solutions: Solutions: E736
- Erdős, Moser
- 1947
(Show Context)
Citation Context ...ution. Many mathematicians realized that while the classical bound in Theorem 3.2 is sharp as stated, it can be improved significantly under additional assumptions on v. For instance, Erdős and Moser =-=[14]-=- showed that if the vi are distinct, then pv = O(n −3/2 ln n). They conjectured that the logarithmic term is not necessary and this was confirmed by Sárközy and Szemerédi [39]. Again, the bound is sha... |

1 |
On Wigner’s semi-cirle law for the eigenvalues of random matrices
- Arnold
- 1971
(Show Context)
Citation Context ... other distributions, such as the Bernoulli distribution (in which each xij equals +1 with probability 1/2 and −1 with probability 1/2). His work has been extended and strengthened in several aspects =-=[1, 2, 34]-=-. The most general form was proved by Pastur [34]: 1 We say that a collection µn of probability measures converges to a limit µ if one has ∫ f dµn → ∫ f dµ for every continuous compactly supported fun... |

1 | The ranks of random graphs
- Costello, Vu
(Show Context)
Citation Context ...ut higher degree polynomials ? In [6], it was shown that for a quadratic form Q := ∑ 1≤i,j≤n cijξiξj with non-zero coefficients, P(Q = z) is O(n−1/8 ). It is simple to improve this bound to O(n−1/4 ) =-=[7]-=-. On the other hand, we conjecture that the truth is O(n −1/2 ), which would be sharp by taking Q = (ξ1 + · · · + ξn) 2 . Costello (personal communication) recently improved the bound to O(n −3/8 ), a... |

1 |
Matrix computations, 3rd Edtion
- Golub, Loan
- 1996
(Show Context)
Citation Context ...nvertible, we set κ(M) = ∞.) The condition number plays a crucial role in numerical linear algebra. The accuracy and stability of most algorithms used to solve the equation Mx = b depend on κ(M) (see =-=[5, 23]-=-, for example). The exact solution x = M −1b, in theory, can be computed quickly (by Gaussian elimination, say). However, in practice computers can only represent a finite subset of real numbers and t... |

1 |
On random (−1, 1) matrices: Singularity and determinant
- Tao, Vu
(Show Context)
Citation Context ...and side can be bounded efficiently, thanks to the fact that all di are large with overwhelming probability, which, in turn, is a consequence of Talagrand’s inequality [46]: Lemma 5.8 (Distance Lemma =-=[52, 55]-=-). With probability 1 − n−ω(1) , the √ distance 1 from a random row vector to a subspace of codimension k is at least 100 k/n, as long as k ≫ log n. Thus, with overwhelming probability, ∑m i=1 d−2 i i... |