scholarly journals TENSOR PRODUCTS OF RANDOM UNITARY MATRICES

2012 ◽  
Vol 01 (04) ◽  
pp. 1250009 ◽  
Author(s):  
TOMASZ TKOCZ ◽  
MAREK SMACZYŃSKI ◽  
MAREK KUŚ ◽  
OFER ZEITOUNI ◽  
KAROL ŻYCZKOWSKI
Keyword(s):  
Tensor Product ◽  
Random Matrices ◽  
Tensor Products ◽  
Unitary Matrices ◽  
Product Of Random Matrices ◽  
Spectral Statistics ◽  
Random Unitary Matrices ◽  
Unitary Ensemble

Tensor products of M random unitary matrices of size N from the circular unitary ensemble are investigated. We show that the spectral statistics of the tensor product of random matrices becomes Poissonian if M = 2, N become large or M become large and N = 2.

scholarly journals Thinning and conditioning of the circular unitary ensemble

2017 ◽  
Vol 06 (02) ◽  
pp. 1750007 ◽  
Author(s):  
Christophe Charlier ◽  
Tom Claeys
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Random Matrices ◽  
Unitary Matrices ◽  
Toeplitz Determinants ◽  
Random Unitary Matrices ◽  
Gap Probabilities ◽  
Unitary Ensemble

We apply the operation of random independent thinning on the eigenvalues of [Formula: see text] Haar distributed unitary random matrices. We study gap probabilities for the thinned eigenvalues, and we study the statistics of the eigenvalues of random unitary matrices which are conditioned such that there are no thinned eigenvalues on a given arc of the unit circle. Various probabilistic quantities can be expressed in terms of Toeplitz determinants and orthogonal polynomials on the unit circle, and we use these expressions to obtain asymptotics as [Formula: see text].

scholarly journals Matrix models for $$\varvec{\varepsilon }$$-free independence

2021 ◽  
Author(s):  
Ian Charlesworth ◽  
Benoît Collins
Keyword(s):  
Tensor Product ◽  
Matrix Models ◽  
Random Matrices ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Product Structure ◽  
Graph Products ◽  
Products Of Random Matrices ◽  
Von Neumann ◽  
Free Independence

AbstractWe investigate tensor products of random matrices, and show that independence of entries leads asymptotically to $$\varepsilon $$ ε -free independence, a mixture of classical and free independence studied by Młotkowski and by Speicher and Wysoczański. The particular $$\varepsilon $$ ε arising is prescribed by the tensor product structure chosen, and conversely, we show that with suitable choices an arbitrary $$\varepsilon $$ ε may be realized in this way. As a result, we obtain a new proof that $$\mathcal {R}^\omega $$ R ω -embeddability is preserved under graph products of von Neumann algebras, along with an explicit recipe for constructing matrix models.

scholarly journals Random Matrices, Magic Squares and Matching Polynomials

10.37236/1859 ◽  
2004 ◽  
Vol 11 (2) ◽  
Author(s):  
Persi Diaconis ◽  
Alex Gamburd
Keyword(s):  
Random Matrices ◽  
Riemann Zeta Function ◽  
Quantum Chaos ◽  
Higher Moments ◽  
Characteristic Polynomials ◽  
Stochastic Matrices ◽  
Unitary Matrices ◽  
Magic Squares ◽  
Riemann Zeta ◽  
Random Unitary Matrices

Characteristic polynomials of random unitary matrices have been intensively studied in recent years: by number theorists in connection with Riemann zeta-function, and by theoretical physicists in connection with Quantum Chaos. In particular, Haake and collaborators have computed the variance of the coefficients of these polynomials and raised the question of computing the higher moments. The answer turns out to be intimately related to counting integer stochastic matrices (magic squares). Similar results are obtained for the moments of secular coefficients of random matrices from orthogonal and symplectic groups. Combinatorial meaning of the moments of the secular coefficients of GUE matrices is also investigated and the connection with matching polynomials is discussed.

scholarly journals Extremal spacings between eigenphases of random unitary matrices and their tensor products

2013 ◽  
Vol 88 (5) ◽  
Author(s):  
Marek Smaczyński ◽  
Tomasz Tkocz ◽  
Marek Kuś ◽  
Karol Życzkowski
Keyword(s):  
Tensor Products ◽  
Unitary Matrices ◽  
Random Unitary Matrices

scholarly journals Eigenvector Convergence for Minors of Unitarily Invariant Infinite Random Matrices

2020 ◽  
Author(s):  
Joseph Najnudel
Keyword(s):  
Weak Convergence ◽  
Random Matrices ◽  
Invariant Measures ◽  
Alternative Proof ◽  
Probability Measures ◽  
Cayley Transform ◽  
Hermitian Matrices ◽  
Unitary Matrices ◽  
Extreme Eigenvalues ◽  
Unitary Ensemble

Abstract In [10], Pickrell fully characterizes the unitarily invariant probability measures on infinite Hermitian matrices. An alternative proof of this classification is given by Olshanski and Vershik in [9], and in [3] Borodin and Olshanski deduce from this proof that under any of these invariant measures, the extreme eigenvalues of the minors, divided by the dimension, converge almost surely. In this paper, we prove that one also has a weak convergence for the eigenvectors, in a sense that is made precise. After mapping Hermitian to unitary matrices via the Cayley transform, our result extends a convergence proven in our paper with Maples and Nikeghbali [6], for which a coupling of the circular unitary ensemble of all dimensions is considered.

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

scholarly journals SUBPROJECTIVITY OF PROJECTIVE TENSOR PRODUCTS OF BANACH SPACES OF CONTINUOUS FUNCTIONS

2021 ◽  
pp. 1-14
Author(s):  
R.M. CAUSEY
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Tensor Products ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Spaces Of Continuous Functions

Abstract Galego and Samuel showed that if K, L are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is c0-saturated if and only if it is subprojective if and only if K and L are both scattered. We remove the hypothesis of metrizability from their result and extend it from the case of the twofold projective tensor product to the general n-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces K1, …, K n , $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is c0-saturated if and only if it is subprojective if and only if each K i is scattered.

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