## Asymptotic freeness almost everywhere for random matrices (2000)

Venue: | Acta Sci. Math. (Szeged |

Citations: | 18 - 0 self |

### BibTeX

@ARTICLE{Hiai00asymptoticfreeness,

author = {Fumio Hiai and Dénes Petz},

title = {Asymptotic freeness almost everywhere for random matrices},

journal = {Acta Sci. Math. (Szeged},

year = {2000},

pages = {801--826}

}

### Years of Citing Articles

### OpenURL

### Abstract

Voiculescu’s asymptotic freeness result for random matrices is improved to the sense of almost everywhere convergence. The asymptotic freeness almost everywhere is first shown for standard unitary matrices based on the computation of multiple moments of their entries, and then it is shown for rather general unitarily invariant selfadjoint random matrices (in particular, standard selfadjoint Gaussian matrices) by applying the first result to the unitary parts of their diagonalization. Bi-unitarily invariant non-selfadjoint random matrices are also treated via polar decomposition.

### Citations

667 |
Random Matrices
- Mehta
- 1991
(Show Context)
Citation Context ...dius) of symmetric or non-symmetric random matrices (see [7], [8], [3], [4] and also recent [9]). We also know that random unitary matrices sometimes play important roles in random matrix theory (see =-=[13]-=-), and the almost sure limit distribution of standard random unitary matrices is the uniform distribution on the unit circle. The asymptotic free property of random matrices is central in recent break... |

304 |
Free Random Variables
- Voiculescu, Dykema, et al.
- 1992
(Show Context)
Citation Context ...arch (C)09640152. 2 Supported by OTKA F023447 and FKFP 502-121. 1operations) which is a noncommutative probability space with the tracial functional τn (in the terminology in free probability theory =-=[25]-=-). The classical Wigner theorem ([27], [28]) tells us that the mean spectral density of certain random symmetric matrices tends to the semicircle law if the matrix size goes to infinity. This converge... |

187 |
On the distribution of the roots of certain symmetric matrices
- Wigner
- 1958
(Show Context)
Citation Context ...47 and FKFP 502-121. 1operations) which is a noncommutative probability space with the tracial functional τn (in the terminology in free probability theory [25]). The classical Wigner theorem ([27], =-=[28]-=-) tells us that the mean spectral density of certain random symmetric matrices tends to the semicircle law if the matrix size goes to infinity. This convergence is concerned with the eigenvalue distri... |

186 |
Characteristic vectors of bordered matrices with infinite dimensions
- Wigner
- 1955
(Show Context)
Citation Context ... F023447 and FKFP 502-121. 1operations) which is a noncommutative probability space with the tracial functional τn (in the terminology in free probability theory [25]). The classical Wigner theorem (=-=[27]-=-, [28]) tells us that the mean spectral density of certain random symmetric matrices tends to the semicircle law if the matrix size goes to infinity. This convergence is concerned with the eigenvalue ... |

167 | The analogues of entropy and of Fisher’s information measure in free probability theory
- Voiculescu
- 1993
(Show Context)
Citation Context ...nstant matrices in these results has played a crucial role in applications to von Neumann algebra theory (in particular, to problems on free group factors) ([21], [15]–[17], [6]) and to free entropy (=-=[23]-=-, [24]). The paper [18] is concerned with the asymptotic freeness for matrices having bosonic and fermionic creations as entries. Our motivation in the present paper is twofold. On one hand, we want t... |

132 |
Limit laws for random matrices and free products
- Voiculescu
- 1991
(Show Context)
Citation Context ...free relation of noncommutative random variables can be also modeled by random matrix ensembles if the matrix size goes to infinity. The asymptotic freeness result was first established by Voiculescu =-=[22]-=- in the case of Gaussian random matrices together with diagonal constant matrices. Further, Dykema [5] proved the same result in the case of general (non-Gaussian) random matrices together with block-... |

97 |
A limit theorem for the norm of random matrices
- Geman
- 1980
(Show Context)
Citation Context .... Furthermore, several results are known about the almost sure convergence of the largest/smallest eigenvalue or the norm (also the spectral radius) of symmetric or non-symmetric random matrices (see =-=[7]-=-, [8], [3], [4] and also recent [9]). We also know that random unitary matrices sometimes play important roles in random matrix theory (see [13]), and the almost sure limit distribution of standard ra... |

69 |
Limit of the smallest eigenvalue of a large dimensional sample covariance matrix
- Bai, Yin
- 1993
(Show Context)
Citation Context ...several results are known about the almost sure convergence of the largest/smallest eigenvalue or the norm (also the spectral radius) of symmetric or non-symmetric random matrices (see [7], [8], [3], =-=[4]-=- and also recent [9]). We also know that random unitary matrices sometimes play important roles in random matrix theory (see [13]), and the almost sure limit distribution of standard random unitary ma... |

65 | The strong limits of random matrix spectra for sample matrices of independent elements - Wachter - 1978 |

56 | Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group
- Radulescu
- 1994
(Show Context)
Citation Context ...nt matrices. The inclusion of constant matrices in these results has played a crucial role in applications to von Neumann algebra theory (in particular, to problems on free group factors) ([21], [15]–=-=[17]-=-, [6]) and to free entropy ([23], [24]). The paper [18] is concerned with the asymptotic freeness for matrices having bosonic and fermionic creations as entries. Our motivation in the present paper is... |

49 |
A strengthened asymptotic freeness result for random matrices with applications to free entropy
- Voiculescu
- 1998
(Show Context)
Citation Context ...rices. Further, Dykema [5] proved the same result in the case of general (non-Gaussian) random matrices together with block-diagonal constant matrices with bounded block-size, and recently Voiculescu =-=[24]-=- proved his asymptotic freeness result without restriction on the type of constant matrices. The inclusion of constant matrices in these results has played a crucial role in applications to von Neuman... |

48 | S.: Random matrices with complex Gaussian entries
- Haagerup, Thorbjørnsen
- 2003
(Show Context)
Citation Context ...known about the almost sure convergence of the largest/smallest eigenvalue or the norm (also the spectral radius) of symmetric or non-symmetric random matrices (see [7], [8], [3], [4] and also recent =-=[9]-=-). We also know that random unitary matrices sometimes play important roles in random matrix theory (see [13]), and the almost sure limit distribution of standard random unitary matrices is the unifor... |

40 |
On the asymptotic distribution of the eigenvalues of random matrices
- Arnold
- 1967
(Show Context)
Citation Context ...symmetric matrices tends to the semicircle law if the matrix size goes to infinity. This convergence is concerned with the eigenvalue distribution with respect to the functionals τn. Later on, Arnold =-=[1]-=- proved that the empirical spectral density of real symmetric (also complex selfadjoint) random matrices with independent entries converges to the semicircle law almost everywhere, that is, the distri... |

36 |
L.: The distribution of eigenvalues in certain sets of random matrices
- Marchenko, Pastur
- 1967
(Show Context)
Citation Context ...ue density in the almost sure sense. For instance, certain non-selfadjoint random matrices admit the circular law as the limiting eigenvalue density ([10], [2]), and the Marchenko-Pastur distribution =-=[11]-=- appears as the limit distribution of Wishart matrices ([26], [14]). Furthermore, several results are known about the almost sure convergence of the largest/smallest eigenvalue or the norm (also the s... |

35 |
Necessary and sufficient condition for the almost sure convergence of the largest eigenvalue of Wigner Matrices Ann
- Bai, Yin
- 1988
(Show Context)
Citation Context ...ore, several results are known about the almost sure convergence of the largest/smallest eigenvalue or the norm (also the spectral radius) of symmetric or non-symmetric random matrices (see [7], [8], =-=[3]-=-, [4] and also recent [9]). We also know that random unitary matrices sometimes play important roles in random matrix theory (see [13]), and the almost sure limit distribution of standard random unita... |

22 | On certain free product factors via an extended matrix model
- Dykema
- 1993
(Show Context)
Citation Context ...matrix size goes to infinity. The asymptotic freeness result was first established by Voiculescu [22] in the case of Gaussian random matrices together with diagonal constant matrices. Further, Dykema =-=[5]-=- proved the same result in the case of general (non-Gaussian) random matrices together with block-diagonal constant matrices with bounded block-size, and recently Voiculescu [24] proved his asymptotic... |

19 | R-diagonal pairs – a common approach to Haar unitaries and circular elements Fields Institute Communications, Volume 12
- NICA, SPEICHER
- 1997
(Show Context)
Citation Context ...re γn is the Haar measure on U(n). This shows that U is Haar distributed and U, H are independent. Hence the required properties of U, H are shown. The notion of R-diagonal elements was introduced in =-=[12]-=-. In place of the definition we here state its characterization shown in [12], p. 155 as a lemma. Lemma 4.2 Let (A,ϕ) be a C ∗ -probability space such that ϕ is a tracial state. An element x ∈Ais R-di... |

16 | The fundamental group of the von Neumann algebra of a free group with infinitely many generators is R
- Rădulescu
- 1992
(Show Context)
Citation Context ...onstant matrices. The inclusion of constant matrices in these results has played a crucial role in applications to von Neumann algebra theory (in particular, to problems on free group factors) ([21], =-=[15]-=-–[17], [6]) and to free entropy ([23], [24]). The paper [18] is concerned with the asymptotic freeness for matrices having bosonic and fermionic creations as entries. Our motivation in the present pap... |

16 | Mixed moments of Voiculescu’s Gaussian random matrices
- Thorbjørnsen
- 1999
(Show Context)
Citation Context ...rent approach was adopted by Xu [29] to obtain asymptotic freeness results for unitary random matrices. Moreover, almost sure convergence of mixed moments of random matrices was recently discussed in =-=[20]-=- too. In Sect. 1 we start with computation of multiple moments of entries of a standard unitary. A convenient proof of the almost sure convergence of standard selfadjoint Gaussian matrices is also giv... |

15 |
A random matrix model from two-dimensional YangMills theory
- Xu
- 1997
(Show Context)
Citation Context ...is that we can treat unitarily invariant selfadjoint and bi-unitarily invariant non-selfadjoint random matrices more generally than Gaussian matrices. Note that a different approach was adopted by Xu =-=[29]-=- to obtain asymptotic freeness results for unitary random matrices. Moreover, almost sure convergence of mixed moments of random matrices was recently discussed in [20] too. In Sect. 1 we start with c... |

13 |
On the eigenvalue distribution of some symmetric random matrices Acta Sci
- Oravecz, Petz
- 1997
(Show Context)
Citation Context ...fadjoint random matrices admit the circular law as the limiting eigenvalue density ([10], [2]), and the Marchenko-Pastur distribution [11] appears as the limit distribution of Wishart matrices ([26], =-=[14]-=-). Furthermore, several results are known about the almost sure convergence of the largest/smallest eigenvalue or the norm (also the spectral radius) of symmetric or non-symmetric random matrices (see... |

12 |
The spectral radius of large random matrices
- Geman
- 1986
(Show Context)
Citation Context ...thermore, several results are known about the almost sure convergence of the largest/smallest eigenvalue or the norm (also the spectral radius) of symmetric or non-symmetric random matrices (see [7], =-=[8]-=-, [3], [4] and also recent [9]). We also know that random unitary matrices sometimes play important roles in random matrix theory (see [13]), and the almost sure limit distribution of standard random ... |

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 ... particular distributions as the limiting eigenvalue density in the almost sure sense. For instance, certain non-selfadjoint random matrices admit the circular law as the limiting eigenvalue density (=-=[10]-=-, [2]), and the Marchenko-Pastur distribution [11] appears as the limit distribution of Wishart matrices ([26], [14]). Furthermore, several results are known about the almost sure convergence of the l... |

7 |
Free convolution and the random sum of matrices
- Speicher
- 1993
(Show Context)
Citation Context ...matrices. Indeed, in [22] Voiculescu obtained the asymptotic freeness of standard unitaries by taking the unitary parts in the polar decomposition of non-selfadjoint Gaussian matrices. Also, Speicher =-=[19]-=- 2used a similar method to show the almost sure limit spectral density of the sum of two selfadjoint matrices. Our approach is opposite. In this paper we first treat the asymptotic freeness of standa... |

5 |
Limit distributions of matrices with bosonic and fermionic entries. In Free probability theory
- Shlyakhtenko
- 1995
(Show Context)
Citation Context ...e results has played a crucial role in applications to von Neumann algebra theory (in particular, to problems on free group factors) ([21], [15]–[17], [6]) and to free entropy ([23], [24]). The paper =-=[18]-=- is concerned with the asymptotic freeness for matrices having bosonic and fermionic creations as entries. Our motivation in the present paper is twofold. On one hand, we want to prove the asymptotic ... |

4 | Stable equivalence of the weak closures of free groups convolution algebras - Radulescu |