## Products of random projections, Jacobi ensembles and universality problems arising from free probability. Probab. Theory Rel (2005)

Venue: | Fields |

Citations: | 30 - 1 self |

### BibTeX

@ARTICLE{Collins05productsof,

author = {Benoît Collins},

title = {Products of random projections, Jacobi ensembles and universality problems arising from free probability. Probab. Theory Rel},

journal = {Fields},

year = {2005}

}

### OpenURL

### Abstract

ABSTRACT. We consider the product of two independent randomly rotated projectors. The square of its radial part turns out to be distributed as a Jacobi ensemble. We study its global and local properties in the large dimension scaling relevant to free probability theory. We establish asymptotics for one point and two point correlation functions, as well as properties of largest and smallest eigenvalues. 1. INTRODUCTION. In this paper, we consider the asymptotic distribution of eigenvalues of a random matrix of the form πn ˜πnπn where πn and ˜πn are independent n×n random orthogonal projections, of ranks qn and ˜qn, whose distributions are invariant under unitary conjugation. This question is part of a more general

### Citations

637 | Random Matrices - Mehta - 1991 |

542 |
Orthogonal polynomials
- Szegö
- 1975
(Show Context)
Citation Context ...orm a sequence of orthogonal polynomials with respect to the measure Γ(a+b+2) 2a+b+1 Γ(a+1)Γ(b+1) 1[−1,1]w a,b (x)dx where w a,b (x) = (1 − x) a (1 + x) b The normalization constant is such that (see =-=[Sze75]-=-, Equation (4.3.4)) P a,b ( ) n + a n (1) = n n )n≥0 Remark. Observe that we choose to consider the weight (1 − x) a (1 + x) b instead of (1 − x) a x b in order to respect the conventional notation fo... |

295 |
Free Random Variables
- Voiculescu, Dykema, et al.
- 1992
(Show Context)
Citation Context ...on converging towards µ1 ⊠ µ2, where µ1 is the probability and µ2 is the probability (1 − α)δ0 + αδ1 Let (1 − β)δ0 + βδ1 r± = α + β − 2αβ ± √ 4αβ(1 − α)(1 − β) By a standard S-transform argument (see =-=[VDN92]-=-, example 3.6.7), √ (r+ − x)(x − r−) µ1⊠µ2 = [1−min(α, β)]δ0+[max(α+β−1, 0)]δ1+ 1[r−,r+]dx 2πx(1 − x) By Theorem 2.2, we recover a short proof of the following result:8 BENOÎT COLLINS Proposition 3.2... |

294 |
Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. AMS, 2000 [2] Deift P, Kamvissis S, Kriecherbauer T, Zhou X. The Toda rarefaction problem. Comm Pure Appl Math
- Deift
- 1996
(Show Context)
Citation Context ...ere exists a kernel which we will describe at section 3.3.3, such that: K a,b n P(Λ a,b n ∩ [x1, x1 + dx1] = 1, . . ., Λ a,b n ∩ [x1, x1 + dx1] = 1) = )2 dx1. . .dxn det(K a,b n (xi, xj)) We refer to =-=[Dei99]-=-, and to [Meh91] for a probabilistic interpretation. Furthermore, P(λ1 ≤ x) can be computed explicitely and its value is (see Equation (5.42) p. 114 of [Dei99]) (5) ∞∑ j=0 (−1) j j! ∫ [x,∞] ∫ . . . [x... |

127 | editors. Handbook of mathematical functions - Abramowitz, Stegun - 1965 |

101 | Universality at the edge of the spectrum in Wigner random matrices - Soshnikov - 1999 |

52 |
Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free
- Collins
- 2003
(Show Context)
Citation Context ... in an analogous obvious sense. The following result was contained in [Voi98] and in [Xu97] under slightly stronger hypotheses. For a proof in full generality, see [Col03a], Proposition 2.3.3 p.52 or =-=[Col03b]-=-, Theorem 3.1. Theorem 3.1. Let U1, · · · , Uk, · · · be a collection of independent Haar distributed random matrices of Mn(C) and (W n i )i∈I be a set of constant matrices of Mn(C) admitting a joint ... |

49 | Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices - Johansson |

46 |
A strengthened asymptotic freeness result for random matrices with applications to free entropy
- Voiculescu
- 1998
(Show Context)
Citation Context ...rges in distribution towards a free random variable. Asymptotic freeness of sequence of collections of random variables is defined in an analogous obvious sense. The following result was contained in =-=[Voi98]-=- and in [Xu97] under slightly stronger hypotheses. For a proof in full generality, see [Col03a], Proposition 2.3.3 p.52 or [Col03b], Theorem 3.1. Theorem 3.1. Let U1, · · · , Uk, · · · be a collection... |

34 | Universality for eigenvalue correlations from the modified Jacobi unitary ensemble
- Kuijlaars, Vanlessen
(Show Context)
Citation Context ... works of [Joh01, Sos99] for important breakthroughs towards these conjectures). The universality conjectures at the hard edge in different frameworks have been established by Kuijlaars and Vanlessen =-=[KV02]-=- and our result extends a part of their work without using Riemann-Hilbert methods. As for universality conjectures at the soft edge, a recent work of Ledoux [Led02] gives explicit non asymptotic boun... |

18 |
Finite deFinetti theorem in linear models and multivariate analysis
- Diaconis, Eaton, et al.
- 1992
(Show Context)
Citation Context ...finity such that there exists a C > 0 such that q3 n ≤ Cn. Then | √ n/qnπ ∗ n,qn (µn) − νqn| = o(1)26 BENOÎT COLLINS where | · | denotes the total variation measure. This result was already known to =-=[DEL92]-=- under the assumption that q3 n = o(n). Jiang informed the author that he recently obtained by different methods ([Jia03]) an improvement of this theorem to the case qn = o(n2 ). The Laguerre Polynomi... |

17 | Strong asymptotics and the limit distribution of the zeros of Jacobi polynomials P (an+α,bn+β) n - Gawronski, Shawyer - 1991 |

16 |
Differential operators and spectral distributions of invariant ensembles from the classical orthogonal polynomials part I: the continuous case. Elect
- Ledoux
- 2004
(Show Context)
Citation Context ...ablished by Kuijlaars and Vanlessen [KV02] and our result extends a part of their work without using Riemann-Hilbert methods. As for universality conjectures at the soft edge, a recent work of Ledoux =-=[Led02]-=- gives explicit non asymptotic bound for the tail of the distribution of the largest eigenvalue of a modified Jacobi ensemble. Our cornerstone result is Theorem 2.2: Theorem. Let X, X ′ ∈ Mqn(C) be in... |

16 |
A random matrix model from two-dimensional Yang-Mills theory
- Xu
- 1997
(Show Context)
Citation Context ...ution towards a free random variable. Asymptotic freeness of sequence of collections of random variables is defined in an analogous obvious sense. The following result was contained in [Voi98] and in =-=[Xu97]-=- under slightly stronger hypotheses. For a proof in full generality, see [Col03a], Proposition 2.3.3 p.52 or [Col03b], Theorem 3.1. Theorem 3.1. Let U1, · · · , Uk, · · · be a collection of independen... |

13 |
Maxima of entries of Haar distributed matrices
- Jiang
- 2005
(Show Context)
Citation Context ...re | · | denotes the total variation measure. This result was already known to [DEL92] under the assumption that q3 n = o(n). Jiang informed the author that he recently obtained by different methods (=-=[Jia03]-=-) an improvement of this theorem to the case qn = o(n2 ). The Laguerre Polynomial (La n)n≥0 is a family of orthogonal polynomials with respect to the measure xae−x1[0,∞) such that the leading coeffici... |

13 |
On the zeros of Jacobi polynomials P (αn,βn) n (x
- Moak, Saff, et al.
- 1979
(Show Context)
Citation Context ...ty functions actually holds uniformly on any compact set containing neither r nor s. Remark. The distribution fn(x)dx already appeared in the study of zeros and asymptotics of Jacobi polynomials (see =-=[MSV79]-=-). Theorem 2.2 provides a simple explanation for the apparition of the same distribution in two a priori very different places of mathematics. It is widely believed that “reasonable” unitary ensembles... |

3 | G.I.) Unitary representations of infinite dimensional pairs (G,K) and the formalism of R. Howe. in Representation of Lie groups and related topics, Adv - Ol’shanskij - 1990 |

2 |
Matrix jacobi process
- Doumerc
(Show Context)
Citation Context ...totic freeness for so-called “Beta Matrices” whose eigenvalue distribution actually follows Jacobi ensembles. Theorem 2.2 can also be found under a different formulation and for different purposes in =-=[Dou03]-=-. To the knowledge of the author, the link between products of randomly rotated projections and Jacobi ensembles had only been observed asymptotically so far, and not at the finite dimension level. In... |

1 |
Log-gases and Random matrices, Chapter 2
- Forrester
(Show Context)
Citation Context ...ually we will see that the random matrix πn ˜πnπn is distributed according to a Jacobi ensemble of parameters (qn, n − ˜qn − qn, ˜qn − qn). For the definition, see Equation (1) and for a good review, =-=[For02]-=-. We use asymptotic properties of Jacobi polynomials in order to derive the asymptotic distributions of eigenvalues. We find that the one point function has an explicit limit, which we relate to free ... |

1 |
Lectures notes of the free probability semester at IHP
- Nica, Speicher
- 2000
(Show Context)
Citation Context ...) converges towards the so-called Marcenko-Pastur distribution √ (u − x)(x − v) const. 1[u,v]dx x where u = 2 + α − 2 √ 1 + α, v = 2 + α + 2 √ 1 + α With Speicher’s non-crossing cumulants theory (see =-=[NS00]-=-), one can prove that this distribution is both a free chi-square distribution and a free Poisson distribution. Upon knowing that the average eigenvalue counting measure of the GUE converges towards t... |