Polynomial with cyclic monotone elements with applications to Random Matrices with discrete spectrum

Arizmendi, Octavio; Celestino, Adrian

doi:10.1142/s2010326321500209

Cited by 5 publications

(9 citation statements)

References 3 publications

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“…One direct corollary of Theorem 3.5 is the asymptotic cyclic monotone independence of [CHS18], stated as follows (see also [AC21]).…”

Section: Theorem 36 (Corollary 17 Of [Mm13a]) Let F Be a Continuous R...mentioning

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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“…One direct corollary of Theorem 3.5 is the asymptotic cyclic monotone independence of [CHS18], stated as follows (see also [AC21]).…”

Section: Theorem 36 (Corollary 17 Of [Mm13a]) Let F Be a Continuous R...mentioning

confidence: 99%

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

Cébron¹,

Dahlqvist²,

Gabriel³

2022

Preprint

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“…First observe that the trivial example ω = 0 shows that some conditions are necessary. Indeed, if ω = 0 and a, b ∈ A are self-adjoint then ω((a (1) + b (2) ) 2 ) = 2ϕ(a)ϕ(b), which can be negative, although both ϕ and ω are positive.…”

Section: Remark 54 (Positivity) It Is Not Clear Under What Conditions...mentioning

confidence: 99%

“…Remark 7.5. Cyclic-monotone independence already appeared in the random matrix model in [5] (see also [20,2] and Section 1), where independence was defined for a pair of * -subalgebras and only for ω.…”

Section: Cyclic-boolean Infinite Divisibilitymentioning

confidence: 99%

Cyclic independence: Boolean and monotone

Arizmendi¹,

Hasebe²,

Lehner³

2022

Preprint

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The present paper introduces a modified version of cyclic-monotone independence which originally arose in the context of random matrices, and also introduces its natural analogy called cyclic-Boolean independence. We investigate formulas for convolutions, limit theorems for sums of independent random variables, and also classify infinitely divisible distributions with respect to cyclic-Boolean convolution. Finally, we provide applications to the eigenvalues of the adjacency matrices of iterated star products of graphs and also iterated comb products of graphs.

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“…where A i " A pN q i and F i " F pN q i are deterministic matrices in M N pCq, F i are of uniformly bounded ranks and tU 1 , U 2 , V 1 , V 2 u are independent Haar unitary matrices. 2 We assume that, in the large N limit, the "main parts" U i A i U i converge in distribution (with respect to the normalized trace) to elements a i in a unital algebra A, and the "perturbation parts" F i converge in distribution (with respect to the nonnormalized trace) to elements f i in an A-algebra F. The sums U i A i U i `Fi then converge (in a suitable sense) to a i `fi P A ' F, which we also denote by pa i , f i q to distinguish the main part and perturbation. The relationship between a i and f i is described in a way quite similar to type B free probability (in the sense of Biane, Goodman and Nica [14]) but slightly different.…”

Section: Overview Of Main Results and Structure Of The Papermentioning

confidence: 99%

“…Although work on outliers of perturbed models is mainly devoted to sums and multiplications, some results on polynomials on unitarily invariant random matrices with perturbation are obtained by Belinschi, Bercovici and Capitaine [7]. For models having only discrete spectra, Arizmendi and Celestino gave an algorithm for computing eigenvalues of polynomials on asymptotically cyclic-monotone independent random matrices [2], and Collins et al gave an algorithm for computing fluctuations of eigenvalues of polynomials on trivially independent 1 random matrices [19].…”

Section: Backgroundsmentioning

confidence: 99%

The Spectra of Principal Submatrices in Rotationally Invariant Hermitian Random Matrices and the Markov– Krein Correspondence

Fujie¹

2022

ALEA

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We prove a concentration phenomenon on the empirical eigenvalue distribution (EED) of the principal submatrix in a random hermitian matrix whose distribution is invariant under unitary conjugacy; for example, this class includes GUE (Gaussian Unitary Ensemble) and Wishart matrices. More precisely, if the EED of the whole matrix converges to some deterministic probability measure m, then the difference of rescaled EEDs of the whole matrix and of its principal submatrix concentrates at the Rayleigh measure (in general, a Schwartz distribution) associated with m by the Markov-Krein correspondence. For the proof, we use the moment method with Weingarten calculus and free probability. At some stage of calculations, the proof requires a relation between the moments of the Rayleigh measure and free cumulants of m. This formula is more or less known, but we provide a different proof by observing a combinatorial structure of non-crossing partitions.

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Polynomial with cyclic monotone elements with applications to Random Matrices with discrete spectrum

Abstract: We provide a generalization and new proofs of the formulas of Collins et al. for the spectrum of polynomials in cyclic monotone elements. This is applied to Random Matrices with discrete spectrum.

Cited by 5 publications

References 3 publications

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

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

Cyclic independence: Boolean and monotone

The Spectra of Principal Submatrices in Rotationally Invariant Hermitian Random Matrices and the Markov– Krein Correspondence

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