Fluctuations and correlations for products of real asymmetric random matrices

FitzGerald, Will; Simm, Nick

doi:10.48550/arxiv.2109.00322

Cited by 5 publications

(8 citation statements)

References 35 publications

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“…This was proven by Forrester [11] and this property holds beyond multiplying Gaussian Ginibre ensembles [19,33,34]. Both the expected number of real eigenvalues and its variance were determined very recently by Simm and his coworker [9,35], as we will recall below, including the opposite limit with N → ∞ at m fixed. Similar considerations have been made very recently for products of truncated orthogonal matrices [26].…”

Section: Introduction and Discussion Of Main Resultsmentioning

confidence: 75%

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: 75%

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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“…Here we see that for general linear statistic the variance diverges (this is in distinction to the case of smooth statistics for GinUE; recall [38, §3.3]), and furthermore the correlations are no longer determinantal but rather Pfaffian. Nonetheless, it has been shown in [159,71] that it can be proved that the higher order cumulants tend to zero as N → ∞, implying the sought central limit theorem. The conclusion is then that the statistic (F − F )/ E r N tends to a zero mean Gaussian, with variance given by the RHS of (2.54) with the factor E r N omitted.…”

Section: Correlation Functionsmentioning

confidence: 99%

“…This gives the result for f at least piecewise continuous. As noted in [71], following an idea in [124], taking advantage of the translation invariance of the limiting two-point truncated correlation function allows the integral instead to be computed in terms of Fourier transforms, and so further enlarging the class of f for its applicability.…”

Section: Correlation Functionsmentioning

confidence: 99%

“…The structural form (2.41) for S r,M N (x, y) also has consequences for the analysis of the two-point correlation and associated fluctuation formula. In particular, it is used in [71] to establish that the structural formula (2.54) for the variance of a linear statistic as N → ∞ holds true independent of M. For the particular linear statistic corresponding to the counting function for the number of real eigenvalues, this formula was shown to break down in the case that M simultaneously tends to ∞ with N. Rather then (σ r,M N ) 2 /E r,M N | M=αN tends to an α dependent quantity; see [5, Th. 1.2].…”

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confidence: 99%

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Progress on the study of the Ginibre ensembles II: GinOE and GinSE

Byun¹,

Forrester²

2023

Preprint

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This is part II of a review relating to the three classes of random non-Hermitian Gaussian matrices introduced by Ginibre in 1965. While part I restricted attention to the GinUE (Ginibre unitary ensemble) case of complex elements, in this part the cases of real elements (GinOE, denoting Ginibre orthogonal ensemble) and quaternion elements represented as 2 × 2 complex blocks (GinSE, denoting Ginibre symplectic ensemble) are considered. The eigenvalues of both GinOE and GinSE form Pfaffian point processes, which are more complicated than the determinantal point processes resulting from GinUE. Nevertheless, many of the obstacles that have slowed progress on the development of traditional aspects of the theory have now been overcome, while new theoretical aspects and new applications have been identified. This permits a comprehensive account of themes addressed too in the complex case: eigenvalue probability density functions and correlation functions, limit formulas for correlation functions, fluctuation formulas, sum rules, gap probabilities and eigenvector statistics, among others. Distinct from the complex case is the need to develop a theory of skew orthogonal polynomials corresponding to the skew inner product associated with the Pfaffian. Another distinct theme is the statistics of real eigenvalues, which is unique to GinOE. These appear in a number of applications of the theory, coming from areas as diverse as diffusion processes and persistence in statistical physics, topologically driven parametric energy level crossings for certain quantum dots, and equilibria counting for a system of random nonlinear differential equations.

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“…See also [110,111] for information about fluctuations and universality properties in such matrix product ensembles, or [112][113][114][115][116] for results concerning their asymptotic density of eigenvalues and singular values (instead of the j.p.d.f.). Using this body of work we can go further in explicitating Conjecture 4 in these known cases.…”

Section: 4mentioning

confidence: 99%

Statistical limits of dictionary learning: random matrix theory and the spectral replica method

Barbier¹,

Macris²

2021

Preprint

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We consider increasingly complex models of matrix denoising and dictionary learning in the Bayes-optimal setting, in the challenging regime where the matrices to infer have a rank growing linearly with the system size. This is in contrast with most existing literature concerned with the low-rank (i.e., constant-rank) regime. We first consider a class of rotationally invariant matrix denoising problems whose mutual information and minimum mean-square error are computable using standard techniques from random matrix theory. Next, we analyze the more challenging models of dictionary learning. To do so we introduce a novel combination of the replica method from statistical mechanics together with random matrix theory, coined spectral replica method. It allows us to conjecture variational formulas for the mutual information between hidden representations and the noisy data as well as for the overlaps quantifying the optimal reconstruction error. The proposed methods reduce the number of degrees of freedom from Θ(N 2 ) (matrix entries) to Θ(N ) (eigenvalues or singular values), and yield Coulomb gas representations of the mutual information which are reminiscent of matrix models in physics. The main ingredients are the use of HarishChandra-Itzykson-Zuber spherical integrals combined with a new replica symmetric decoupling ansatz at the level of the probability distributions of eigenvalues (or singular values) of certain overlap matrices.

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Fluctuations and correlations for products of real asymmetric random matrices

Cited by 5 publications

References 35 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 $m=O(N)$

Progress on the study of the Ginibre ensembles II: GinOE and GinSE

Statistical limits of dictionary learning: random matrix theory and the spectral replica method

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