Lyapunov exponent, universality and phase transition for products of random matrices

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

doi:10.48550/arxiv.1810.00433

Cited by 14 publications

(35 citation statements)

References 63 publications

(88 reference statements)

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“…It was therefore natural to ask about the existence of a critical regime, that interpolates between such a random matrix behaviour when N → ∞ at m fixed, and the deterministic Lyapunov spectrum when m → ∞ at N fixed [10,22,23,31]. An interpolating kernel was indeed identified [2,3,27] that describes the local statistics in a critical regime, when the two parameters are proportional, m = αN with α > 0 fixed, as conjectured in [1].…”

Section: Introduction and Discussion Of Main Resultsmentioning

confidence: 97%

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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Section: Introduction and Discussion Of Main Resultsmentioning

confidence: 97%

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 are concerned with the asymptotic regime in which both the matrix size and the number of factors in the product approach infinity in such a way that their ratio converges to a positive number. The study of this regime was initiated by Akemann, Burda, and Kieburg in [3,4]; the scaling limit of the corresponding kernel was obtained by Liu, Wang, and Wang [33]. Following the latter paper, we refer to this scaling limit as to the critical kernel and to the corresponding process as to the critical determinantal process.…”

Section: Introductionmentioning

confidence: 95%

“…In Section 2, we describe determinantal processes related to products of random matrices. In particular, the critical determinantal process is defined and the formula for the critical kernel is presented, as stated in Liu, Wang, and Wang [33]. In Section 3, we formulate Proposition 3.1 and Theorem 3.4.…”

Section: Below)mentioning

confidence: 99%

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Gap probability for products of random matrices in the critical regime

Berezin,

Strahov

2021

Preprint

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The singular values of a product of M independent Ginibre matrices of size N ×N form a determinantal point process. As both M and N go to infinity in such a way that M/N → α, α > 0, a scaling limit emerges. We consider a gap probability for the corresponding limiting determinantal process, namely, the probability that there are no particles in the interval (a, +∞), a > 0. This probability is evaluated explicitly in terms of the unique solution of a certain matrix Riemann-Hilbert problem of size 2 × 2. The right-tail asymptotics of this solution is obtained by the Deift-Zhou non-linear steepest descent analysis.

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“…the large n limits of Lyapunov exponents. The regime where T /n converges to a constant was also studied in [3,4,18]. Though the Lyapunov exponents do not directly appear in this regime, it is known from the work of the first author [1] that the asymptotic behavior of the largest singular values for products of Ginibre matrices coincides with that of products of corners of Haar unitary matrices.…”

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confidence: 99%

Lyapunov exponents for truncated unitary and Ginibre matrices

Ahn¹,

Peski²

2021

Preprint

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In this note, we show that the Lyapunov exponents of mixed products of random truncated Haar unitary and complex Ginibre matrices are asymptotically given by equally spaced 'picket-fence' statistics. We discuss how these statistics should originate from the connection between random matrix products and multiplicative Brownian motion on GLn(C), analogous to the connection between discrete random walks and ordinary Brownian motion. Our methods are based on contour integral formulas for products of classical matrix ensembles from integrable probability.

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Lyapunov exponent, universality and phase transition for products of random matrices

Cited by 14 publications

References 63 publications

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

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

Gap probability for products of random matrices in the critical regime

Lyapunov exponents for truncated unitary and Ginibre matrices

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