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Asymptotic Freeness of Unitary Matrices in Tensor Product Spaces for Invariant States
Probability · Submitted 18 Nov 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Here, we extend this result by proving that asymptotic freeness of tensor products of Haar unitary matrices holds with respect to a significantly larger class of states.
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The abstract, sentence by sentence
In this paper, we pursue our study of asymptotic properties of families of random matrices that have a tensor structure. In previous work, the first- and second-named authors provided conditions under which tensor products of unitary random matrices are asymptotically free with respect to the normalized trace. Here, we extend this result by proving that asymptotic freeness of tensor products of Haar unitary matrices holds with respect to a significantly larger class of states. Our result relies on invariance under the symmetric group, and therefore on traffic probability. As a byproduct, we explore two additional generalisations: (i) we state results of freeness in a context of general sequences of representations of the unitary group -- the fundamental representation being a particular case that corresponds to the classical asymptotic freeness result for Haar unitary matrices, and (ii) we consider actions of the symmetric group and the free group simultaneously and obtain a result of asymptotic freeness in this context as well.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper, we pursue our study of asymptotic properties of families of random matrices that have a tensor structure. | background 0.78 | 0.66 | 0.03 | asserted 0.59 |
| s1 | In previous work, the first- and second-named authors provided conditions under which tensor products of unitary random matrices are asymptotically free with respect to the normalized trace. | background 1.00 | 0.13 | 0.03 | asserted 1.00 |
| s2 | Here, we extend this result by proving that asymptotic freeness of tensor products of Haar unitary matrices holds with respect to a significantly larger class of states. | result 0.50 | 0.94 | 0.91 | asserted 1.00 |
| s3 | Our result relies on invariance under the symmetric group, and therefore on traffic probability. | method 0.99 | 0.74 | 0.06 | asserted 1.00 |
| s4 | As a byproduct, we explore two additional generalisations: (i) we state results of freeness in a context of general sequences of representations of the unitary group -- the fundamental representation being a particular case that corresponds to the classical asymptotic freeness result for Haar unitary matrices, and (ii) we consider actions of the symmetric group and the free group simultaneously and obtain a result of asymptotic freeness in this context as well. | contribution 0.77 | 0.94 | 0.09 | asserted 1.00 |
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The paper
Original abstract
In this paper, we pursue our study of asymptotic properties of families of random matrices that have a tensor structure. In previous work, the first- and second-named authors provided conditions under which tensor products of unitary random matrices are asymptotically free with respect to the normalized trace. Here, we extend this result by proving that asymptotic freeness of tensor products of Haar unitary matrices holds with respect to a significantly larger class of states. Our result relies on invariance under the symmetric group, and therefore on traffic probability. As a byproduct, we explore two additional generalisations: (i) we state results of freeness in a context of general sequences of representations of the unitary group -- the fundamental representation being a particular case that corresponds to the classical asymptotic freeness result for Haar unitary matrices, and (ii) we consider actions of the symmetric group and the free group simultaneously and obtain a result of asymptotic freeness in this context as well.
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How the literature uses it
1 arXiv paper cite this in 1 sentence
What it cites · 6 sentences without a confident label
Uses the method or tool of 2 papers
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This part of our argument makes crucial use of a precise asymptotic for the injective trace of Haar unitary matrices from [ref] (see Proposition [ref]).
Traffic Distributions and Independence II: Universal Constructions for Traffic Spaces · Proof Overview Part 2 · 0.90
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Our main tool for this is a splitting lemma that appears in [ref] .
Traffic distributions and independence: permutation invariant random matrices and the three notions of independence · Proof Overview Part 2 · 1.00
Cites as background 1 paper
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In the framework of traffic spaces [ref] , one considers families of random matrices $A_N=(A_j)_j∈J$ such that [equation] is of order 1 for large $N$, where $c(T)$ is the number of connected components of $T$; see for instance Proposition [ref].
Traffic distributions and independence: permutation invariant random matrices and the three notions of independence · Renormalizations · 0.86
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