Phase transitions for infinite products of large non-Hermitian random matrices

Liu, Dang-Zheng; Wang, Yanhui

doi:10.48550/arxiv.1912.11910

Cited by 3 publications

(4 citation statements)

References 64 publications

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“…Do those ensembles yield local spectral statistics that still follow those of Dyson's Brownian motion but now for the GOE and GSE, respectively? It is also interesting to study complex eigenvalues (instead of singular values) of the product of Ginibre matrices using the tools developed in [36], where we refer to [25] for first results. Due to its two-dimensional nature one should expect a different but related behaviour.…”

Section: Discussionmentioning

confidence: 99%

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Universality of local spectral statistics of products of random matrices

Akemann,

Burda,

Kieburg

2020

Preprint

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We derive exact analytical expressions for correlation functions of singular values of the product of M Ginibre matrices of size N in the double scaling limit M, N → ∞. The singular value statistics is described by a determinantal point process with a kernel that interpolates between GUE statistic and Dirac-delta (picket-fence) statistic. In the thermodynamic limit, N → ∞, the interpolation parameter is given by the limiting quotient a = N/M . One of our goals is to find an explicit form of the kernel at the hard edge, in the bulk and at the soft edge for any a. We find that in addition to the standard scaling regimes, there is a new transitional regime which interpolates between the hard edge and the bulk. We conjecture that these results are universal, and that they apply to a broad class of products of random matrices from the Gaussian basin of attraction, including correlated matrices. We corroborate this conjecture by numerical simulations. Additionally, we show that the local spectral statistics of the considered random matrix products is identical with the local statistics of Dyson Brownian motion with the initial condition given by equidistant positions, with the crucial difference that this equivalence holds only locally. Finally, we have identified a mesoscopic spectral scale at the soft edge which is crucial for the unfolding of the spectrum.

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Section: Discussionmentioning

confidence: 99%

“…Furthermore, we present deeper insights into these statistics, including the issue of universality, unfolding, and what they mean. For a first work on the complex eigenvalues statistic, we refer to [25].…”

Section: Introductionmentioning

confidence: 99%

Universality of local spectral statistics of products of random matrices

Akemann,

Burda,

Kieburg

2020

Preprint

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“…In the limit N → ∞ at fixed m, it was shown for products of complex Ginibre matrices that the singular values [28] and complex eigenvalues [29] of the product matrix have the same eigenvalue statistics as a single matrix at m = 1, apart from the origin where the mean density diverges and new universality classes emerge, cf. [4].…”

Section: Introduction and Discussion Of Main Resultsmentioning

confidence: 99%

The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$

Akemann¹,

Byun²

2022

Preprint

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We study the product Pm of m real Ginibre matrices with Gaussian elements of size N , which has received renewed interest recently. Its eigenvalues, which are either real or come in complex conjugate pairs, become all real with probability one when m → ∞ at fixed N . In this regime the statistics becomes deterministic and the Lyapunov spectrum has been derived long ago. On the other hand, when N → ∞ and m is fixed, it can be expected that away from the origin the same local statistics as for a single real Ginibre ensemble at m = 1 prevails. Inspired by analogous findings for products of complex Ginibre matrices, we introduce a critical scaling regime when the two parameters are proportional, m = αN . We derive the expected number, variance and rescaled density of real eigenvalues in this critical regime. This allows us to interpolate between previous recent results in the above mentioned limits when α → ∞ and α → 0, respectively.

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“…The determinantal machinery allowed them to analyse the point process of eigenvalues in various microscopic regimes for fixed m. A similar theory has been developed to analyse products of truncated unitary random matrices [4,1,33]. Recently these developments have been extended to study the question of double scaling limits as both m and N tend to infinity simultaneously [3,31,32].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the number of real eigenvalues of a product of truncated orthogonal random matrices

Little¹,

Mezzadri²,

Simm³

2021

Preprint

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Let O be chosen uniformly at random from the group of (N +L)×(N +L) orthogonal matrices. Denote by Õ the upper-left N × N corner of O, which we refer to as a truncation of O. In this paper we prove two conjectures of Forrester, Ipsen and Kumar [18] on the number of real eigenvalues) ∼ c L,m log(N ) as N → ∞ and compute the constant c L,m explicitly. These results generalise the known results in the one matrix case due to Khoruzhenko, Sommers and Życzkowski [28].

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Phase transitions for infinite products of large non-Hermitian random matrices

Cited by 3 publications

References 64 publications

Universality of local spectral statistics of products of random matrices

Universality of local spectral statistics of products of random matrices

The Product of $m$ real $N\times N$ Ginibre matrices: Real eigenvalues in the critical regime $m=O(N)$

On the number of real eigenvalues of a product of truncated orthogonal random matrices

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