Operator-Valued Free Probability Theory and Block Random Matrices

Mingo, James A.; Speicher, Roland

doi:10.1007/978-1-4939-6942-5_9

Cited by 70 publications

(138 citation statements)

References 11 publications

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“…As the sum of m independent random Wishart matrices, we may find the eigenvalue density of M using the methods of free probability theory. This requires first computing the self-energy Σ (known as the R-transform in free probability theory, see Theorem 18 of Ref [110]) of the single Wishart matrix L † L (which we do not normalize by N ). By inverting the resolvent Eq.…”

Section: Dissipators With Multiple Jump Operatorsmentioning

confidence: 99%

Random Lindblad dynamics

Can

2019

J. Phys. A: Math. Theor.

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We study the mixing behavior of random Lindblad generators with no symmetries, using the dynamical map or propagator of the dissipative evolution. In particular, we determine the longtime behavior of a dissipative form factor, which is the trace of the propagator, and use this as a diagnostic for the existence or absence of a spectral gap in the distribution of eigenvalues of the Lindblad generator. We find that simple generators with a single jump operator are slowly mixing, and relax algebraically in time, due to the closing of the spectral gap in the thermodynamic limit. Introducing additional jump operators or a Hamiltonian opens up a spectral gap which remains finite in the thermodynamic limit, leading to exponential relaxation and thus rapid mixing. We use the method of moments and introduce a novel diagrammatic expansion to determine exactly the form factor to leading order in Hilbert space dimension N . We also present numerical support for our main results.

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Section: Dissipators With Multiple Jump Operatorsmentioning

confidence: 99%

Random Lindblad dynamics

Can

2019

J. Phys. A: Math. Theor.

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“…N is required to be comparable to T [29]. If N > T or T >> N , the effect of our method will be unsatisfactory, which will be clearly discussed in case studies.…”

Section: E Boundary Conditionmentioning

confidence: 97%

“…Theorem 2: Stieltjes Transform One of the most general resolvent of the covariance matrix E is: G E (s) = (sI N − E) −1 (30) where s ∈ C + is a complex variable, and I N ∈ R N ×N is the unit matrix. The Stieltjes transform is defined as [29]:…”

Section: Appendix a Random Matrices Basicsmentioning

confidence: 99%

Improving Power System State Estimation Based on Matrix-Level Cleaning

Yang

Qiu

Chu

et al. 2020

IEEE Trans. Power Syst.

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Power system state estimation is heavily subjected to measurement error, which comes from the noise of measuring instruments, communication noise, and some unclear randomness. Traditional weighted least square (WLS), as the most universal state estimation method, attempts to minimize the residual between measurements and the estimation of measured variables, but it is unable to handle the measurement error. To solve this problem, based on random matrix theory, this paper proposes a data-driven approach to clean measurement error in matrix-level. Our method significantly reduces the negative effect of measurement error, and conducts a two-stage state estimation scheme combined with WLS. In this method, a Hermitian matrix is constructed to establish an invertible relationship between the eigenvalues of measurements and their covariance matrix. Random matrix tools, combined with an optimization scheme, are used to clean measurement error by shrinking the eigenvalues of the covariance matrix. With great robustness and generality, our approach is particularly suitable for large interconnected power grids. Our method has been numerically evaluated using different testing systems, multiple models of measured noise and matrix size ratios.

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“…First, we summarize some definitions from operator algebras and free probability theory. See [9] for the detail. x k µ a (dx) = τ (a k ), k ∈ N.…”

Section: Freenessmentioning

confidence: 99%

“…d (A) be a matrix of * -free standard circular elements. By[8, Proposition 5.30], Λ(C) is a C 2 -valued semicircular element (see[9, Section 9.1] for the definition) with the following variance mapping η∶ C 2 → C 2 :…”

mentioning

confidence: 99%