Lyapunov Exponent, Universality and Phase Transition for Products of Random Matrices

Liu, Dang-Zheng; Wang, Dong; Wang, Yanhui

doi:10.1007/s00220-022-04584-7

Cited by 5 publications

(5 citation statements)

References 44 publications

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“…This is the kernel describing the Lyapunov exponents in the universal regime Γt = O(n). A similar kernel was obtained in formula (1.15) in [100] and in formula VI.22 in [101] in the context of the universal edge statistics of products of Ginibre matrices. The connection to the Dyson Brownian motion was also discussed in [101].…”

Section: Conflicts Of Interestsupporting

confidence: 62%

A Dyson Brownian Motion Model for Weak Measurements in Chaotic Quantum Systems

Gerbino,

Le Doussal,

Giachetti

et al. 2024

Quantum Reports

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We consider a toy model for the study of monitored dynamics in many-body quantum systems. We study the stochastic Schrödinger equation resulting from continuous monitoring with a rate Γ of a random Hermitian operator, drawn from the Gaussian unitary ensemble (GUE) at every time t. Due to invariance by unitary transformations, the dynamics of the eigenvalues {λα}α=1n of the density matrix decouples from that of the eigenvectors, and is exactly described by stochastic equations that we derive. We consider two regimes: in the presence of an extra dephasing term, which can be generated by imperfect quantum measurements, the density matrix has a stationary distribution, and we show that in the limit of large size n→∞ it matches with the inverse-Marchenko–Pastur distribution. In the case of perfect measurements, instead, purification eventually occurs and we focus on finite-time dynamics. In this case, remarkably, we find an exact solution for the joint probability distribution of λ’s at each time t and for each size n. Two relevant regimes emerge: at short times tΓ=O(1), the spectrum is in a Coulomb gas regime, with a well-defined continuous spectral distribution in the n→∞ limit. In that case, all moments of the density matrix become self-averaging and it is possible to exactly characterize the entanglement spectrum. In the limit of large times tΓ=O(n), one enters instead a regime in which the eigenvalues are exponentially separated log(λα/λβ)=O(Γt/n), but fluctuations ∼O(Γt/n) play an essential role. We are still able to characterize the asymptotic behaviors of the entanglement entropy in this regime.

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Section: Conflicts Of Interestsupporting

confidence: 62%

A Dyson Brownian Motion Model for Weak Measurements in Chaotic Quantum Systems

Gerbino,

Le Doussal,

Giachetti

et al. 2024

Quantum Reports

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show abstract

“…In the intermediate regime where τ, N → ∞ and τ /N converges to a constant, a new universal limit object appears. This distribution interpolates between the two aforementioned ones; it first appeared for matrices with iid Gaussian entries in work of Akemann-Burda-Kieburg [6,7] in the physics literature, and in the mathematics literature was shown by Liu-Wang-Wang [61].…”

supporting

confidence: 53%

“…We have already mentioned that in the complex setting, the bulk limits of singular values is governed by the interpolating kernel of [6] rather than the sine kernel. The same is true on the soft edge, namely there is another limit (also introduced in [6] and also treated in the mathematical literature in [61]) which interpolates between the Airy process and independent Gaussian statistics; we mention also work of Berezin-Strahov [10], which computed the gap probabilities of the limit. This soft edge limit was further shown by Ahn for truncated unitary matrices [2] and later for a broad class of invariant ensembles [1].…”

Section: Define Stopping Timesmentioning

confidence: 86%

“…They have since appeared in the study of disordered systems in statistical physics, dynamical systems, convolutional neural networks, and wireless communication channels; see for instance Crisanti-Paladin-Vulpiani [33], Cohen-Newman [32], Hanin-Nica [45], and Akemann-Kieburg-Wei [8] respectively, and the references therein. When N is fixed and τ → ∞, the logarithms of the singular values exhibit independent Gaussian fluctuations in the limit, as shown for the largest singular value by Furstenberg-Kesten [42] and for all by Akemann-Burda-Kieburg [6,7] and Liu-Wang-Wang [61]. In the opposite regime when τ is fixed and N → ∞, Liu-Wang-Zhang [62] showed that the singular values in the bulk are governed by the sine kernel as with eigenvalues of a single matrix.…”

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confidence: 81%

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Limits and fluctuations of p-adic random matrix products

Peski¹

2021

Sel. Math. New Ser.

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We study the distribution of singular numbers of products of certain classes of padic random matrices, as both the matrix size and number of products go to ∞ simultaneously. In this limit, we prove convergence of the local statistics to a new random point configuration on Z, defined explicitly in terms of certain intricate mixed q-series/exponential sums. This object may be viewed as a nontrivial p-adic analogue of the interpolating distributions of Akemann-Burda-Kieburg [6], which generalize the sine and Airy kernels and govern limits of complex matrix products. Our proof uses new Macdonald process computations and holds for matrices with iid additive Haar entries, corners of Haar matrices from GLN (Zp), and the p-adic analogue of Dyson Brownian motion studied in [80]. Contents 1. Introduction 1 2. p-adic matrix background 3. Symmetric function background 4. The limit of the Plancherel/principal Hall-Littlewood measure 5. Residue expansions 6. Tightness and the limiting random variable 7. An indeterminate moment problem 8. Examples of residue formula for L k,t,χ 9. The case of pure α specializations 10. From processes of infinitely many particles to finite matrix bulk limits Appendix A. Parallels with complex matrix products Appendix B. A Hall-Littlewood proof of [80, Theorem 1.2] References

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“…• The double-scaling, or critical, regime, in which the depth L and widths n ℓ tend jointly to infinity [7,8,11,15,[27][28][29].…”

Section: Applications Of Wishart Product Matrices In Science and Tech...mentioning

confidence: 99%

Replica method for eigenvalues of real Wishart product matrices

Zavatone-Veth

Pehlevan

2023

SciPost Phys. Core

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We show how the replica method can be used to compute the asymptotic eigenvalue spectrum of a real Wishart product matrix. For unstructured factors, this provides a compact, elementary derivation of a polynomial condition on the Stieltjes transform first proved by Müller [IEEE Trans. Inf. Theory. 48, 2086-2091 (2002)]. We then show how this computation can be extended to ensembles where the factors are drawn from matrix Gaussian distributions with general correlation structure. For both unstructured and structured ensembles, we derive polynomial conditions on the average values of the minimum and maximum eigenvalues, which in the unstructured case match the results obtained by Akemann, Ipsen, and Kieburg [Phys. Rev. E 88, 052118 (2013)] for the complex Wishart product ensemble.

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Lyapunov Exponent, Universality and Phase Transition for Products of Random Matrices

Cited by 5 publications

References 44 publications

A Dyson Brownian Motion Model for Weak Measurements in Chaotic Quantum Systems

A Dyson Brownian Motion Model for Weak Measurements in Chaotic Quantum Systems

Limits and fluctuations of p-adic random matrix products

Replica method for eigenvalues of real Wishart product matrices

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