<scp>Finite‐Rank</scp> Perturbations of Random Band Matrices via Infinitesimal Free Probability

Au, Benson

doi:10.1002/cpa.21924

Cited by 7 publications

(2 citation statements)

References 49 publications

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“…In [Shl18], using a different line of arguments, including concentration results on large unitary groups, Shlyakhtenko gave a short elegant proof of asymptotic infinitesimal freeness in expectation of a family of unitarily invariant matrices from a tuple of finite rank matrices, and, proved besides, with explicit Gaussian computations, the asymptotic infinitesimal freeness in expectation of a family of independent GUE matrices from a tuple of finite rank matrices. This result has been generalized in three directions: Collins, Hasebe and Sakuma proved in [CHS18] the almost sure asymptotic infinitesimal freeness of a family of unitarily invariant matrices from a tuple of finite rank matrices, which can also be stated asymptotically as cyclic monotone independence; Dallaporta and Février proved in [DF19, Appendix A] that a family of independent GUE matrices is asymptotically infinitesimal free in expectation from a tuple of bounded deterministic matrices converging in * -distribution; and Au proved in [Au21] the asymptotic infinitesimal freeness in expectation of a family of Wigner matrices with finite moments from a tuple of finite rank matrices, and of a family of periodically banded GUE matrix from a tuple of finite rank matrices. Finally, the asymptotic infinitesimal freeness in expectation of two independent random matrices, at least one of them being unitarily invariant, is also a consequence of the general theory of surfaced free probability of Borot, Charbonnier, Garcia-Failde, Leid and Shadrin in [BCGF + 21] (it corresponds to ( 1 2 , 1)-freeness in [BCGF + 21, Theorem 4.28]).…”

Section: Introduction 1infinitesimal Freeness For Random Matricesmentioning

confidence: 99%

Freeness of type $B$ and conditional freeness for random matrices

Cébron¹,

Dahlqvist²,

Gabriel³

2022

Preprint

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The asymptotic freeness of independent unitarily invariant N × N random matrices holds in expectation up to O(N −2 ). An already known consequence is the infinitesimal freeness in expectation. We put in evidence another consequence for unitarily invariant random matrices: the almost sure asymptotic freeness of type B. As byproducts, we recover the asymptotic cyclic monotonicity, and we get the asymptotic conditional freeness. In particular, the eigenvector empirical spectral distribution of the sum of two randomly rotated random matrices converges towards the conditionally free convolution. We also show new connections between infinitesimal freeness, freeness of type B, conditional freeness, cyclic monotonicity and monotone independence. Finally, we show rigorously that the BBP phase transition for an additive rank-one perturbation of a GUE matrix is a consequence of the asymptotic conditional freeness, and the arguments extend to the study of the outlier eigenvalues of other unitarily invariant ensembles. * G.C. has been partly supported by the ERC advanced grant "Noncommutative distributions in free probability", partly by the Project MESA (ANR-18-CE40-006) and partly by the Project STARS (ANR-20-CE40-0008) of the French National Research Agency. G.C. wishes to thank Roland Speicher, for giving him the opportunity to co-organize the Mini-workshop Notions of freeness in April 2015, during which the authors conceived of the first ideas that led to the present work.† F.G. has been partly supported by the ERC SG CONSTAMIS.

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Section: Introduction 1infinitesimal Freeness For Random Matricesmentioning

confidence: 99%

Freeness of type $B$ and conditional freeness for random matrices

Cébron¹,

Dahlqvist²,

Gabriel³

2022

Preprint

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“…N ) i∈I converge in traffic distribution to a semicircular traffic family of covariance [Au18,Au20] for additional work on Wigner matrices and their generalizations in the traffic framework.…”

Section: Introductionmentioning

confidence: 99%

Semicircular families of general covariance from Wigner matrices with permuted entries

Au¹

2021

Preprint

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Let (σ (i) N ) i∈I be a family of symmetric permutations of the entries of a Wigner matrix W N . We characterize the limiting traffic distribution of the corresponding family of dependent Wigner matrices (W) i∈I in terms of the geometry of the permutations. We also consider the analogous problem for the limiting joint distribution of (WIn particular, we obtain a description in terms of semicircular families with general covariance structures. As a special case, we derive necessary and sufficient conditions for traffic independence as well as sufficient conditions for free independence.

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Semicircular families of general covariance from Wigner matrices with permuted entries

2022

Probab. Theory Relat. Fields

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Let $$(\sigma _N^{(i)})_{i \in I}$$ ( σ N ( i ) ) i ∈ I be a family of symmetric permutations of the entries of a Wigner matrix $${\mathbf {W}}_N$$ W N . We characterize the limiting traffic distribution of the corresponding family of dependent Wigner matrices $$({\mathbf {W}}_N^{\sigma _N^{(i)}})_{i \in I}$$ ( W N σ N ( i ) ) i ∈ I in terms of the geometry of the permutations. We also consider the analogous problem for the limiting joint distribution of $$({\mathbf {W}}_N^{\sigma _N^{(i)}})_{i \in I}$$ ( W N σ N ( i ) ) i ∈ I . In particular, we obtain a description in terms of semicircular families with general covariance structures. As a special case, we derive necessary and sufficient conditions for traffic independence as well as sufficient conditions for free independence.

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Finite‐Rank Perturbations of Random Band Matrices via Infinitesimal Free Probability

Cited by 7 publications

References 49 publications

Freeness of type $B$ and conditional freeness for random matrices

Freeness of type $B$ and conditional freeness for random matrices

Semicircular families of general covariance from Wigner matrices with permuted entries

Semicircular families of general covariance from Wigner matrices with permuted entries

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