Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach

Nowak, Maciej A.; Tarnowski, Wojciech

doi:10.1007/jhep06(2018)152

Cited by 24 publications

(30 citation statements)

References 76 publications

(144 reference statements)

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“…[47] which has created some renewed interest recently, cf. [32,19,50]. In any case a quantum state can be decomposed as |ϕ = i,α G i,α |R A i ⊗|R B α , with coefficients given by a complex N × M matrix G. The density matrix of this state is then given by (1.2)…”

Section: Introductionmentioning

confidence: 99%

Products of random matrices from fixed trace and induced Ginibre ensembles

Akemann¹,

Cikovic²

2018

J. Phys. A: Math. Theor.

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We investigate the microcanonical version of the complex induced Ginibre ensemble, by introducing a fixed trace constraint for its second moment. Like for the canonical Ginibre ensemble, its complex eigenvalues can be interpreted as a two-dimensional Coulomb gas, which are now subject to a constraint and a modified, collective confining potential. Despite the lack of determinantal structure in this fixed trace ensemble, we compute all its density correlation functions at finite matrix size and compare to a fixed trace ensemble of normal matrices, representing a different Coulomb gas. Our main tool of investigation is the Laplace transform, that maps back the fixed trace to the induced Ginibre ensemble. Products of random matrices have been used to study the Lyapunov and stability exponents for chaotic dynamical systems, where the latter are based on the complex eigenvalues of the product matrix. Because little is known about the universality of the eigenvalue distribution of such product matrices, we then study the product of m induced Ginibre matrices with a fixed trace constraint -which are clearly non-Gaussian -and M − m such Ginibre matrices without constraint. Using an m-fold inverse Laplace transform, we obtain a concise result for the spectral density of such a mixed product matrix at finite matrix size, for arbitrary fixed m and M . Very recently local and global universality was proven by the authors and their coworker for a more general, single elliptic fixed trace ensemble in the bulk of the spectrum. Here, we argue that the spectral density of mixed products is in the same universality class as the product of M independent induced Ginibre ensembles. arXiv:1711.05488v3 [math-ph]

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Section: Introductionmentioning

confidence: 99%

Products of random matrices from fixed trace and induced Ginibre ensembles

Akemann¹,

Cikovic²

2018

J. Phys. A: Math. Theor.

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“…In [43] a list of results for the off-diagonal overlap in various ensembles with complex matrix elements is given, like the induced Ginibre ensemble when α ∼ N , the truncated unitary, spherical and product ensemble of two Ginibre matrices. In all cases the numerator of (4.21) gets modified, whereas the quartic repulsion is unchanged.…”

Section: Macroscopic Bulk Limit Of the Overlapsmentioning

confidence: 99%

“…An important ingredient in applications of RMT is the introduction of a time dependence or dynamics on the set of eigenvalues, leading to Dyson's Brownian motion for the classical Hermitian ensembles. In a series of papers, using Green's functions and stochastic evolution equations, it was emphasised that for non-Hermitian random matrices the eigenvector and eigenvalue dynamics no longer decouples [12,13,43], with references to further applications e.g. to neural networks.…”

Section: Introductionmentioning

confidence: 99%

Universal eigenvector correlations in quaternionic Ginibre ensembles

Akemann

Förster

Kieburg

2020

J. Phys. A: Math. Theor.

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Non-Hermitian random matrices enjoy non-trivial correlations in the statistics of their eigenvectors. We study the overlap among left and right eigenvectors in Ginibre ensembles with quaternion valued Gaussian matrix elements. This concept was introduced by Chalker and Mehlig in the complex Ginibre ensemble. Using a Schur decomposition, for harmonic potentials we can express the overlap in terms of complex eigenvalues only, coming in conjugate pairs in this symmetry class. Its expectation value leads to a Pfaffian determinant, for which we explicitly compute the matrix elements for the induced Ginibre ensemble with 2α zero eigenvalues, for finite matrix size N . In the macroscopic large-N limit in the bulk of the spectrum we recover the limiting expressions of the complex Ginibre ensemble for the diagonal and off-diagonal overlap, which are thus universal.

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“…It leads us to the conclusion that the understanding of the temporal evolution of asymmetric neural networks requires considering the entangled dynamics of both eigenvectors and eigenvalues, which might bear consequences for learning and memory processes in these models. Considering the success of FRV analysis in a wide variety of branches disciplines, we hope that the results presented here foster additional application of these ideas in the area of brain sciences.Recently, the two-point function associated with off-diagonal elements of the overlap matrix has become accessible within free probability [35].…”

mentioning

confidence: 99%

“…Recently, the two-point function associated with off-diagonal elements of the overlap matrix has become accessible within free probability [35].…”

mentioning

confidence: 99%

From Synaptic Interactions to Collective Dynamics in Random Neuronal Networks Models: Critical Role of Eigenvectors and Transient Behavior

Gudowska–Nowak

Nowak

Chialvo³

et al. 2020

Neural Computation

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The study of neuronal interactions is currently at the center of several neuroscience big collaborative projects (including the Human Connectome, the Blue Brain, the Brainome, etc.) which attempt to obtain a detailed map of the entire brain matrix. Under certain constraints, mathematical theory can advance predictions of the expected neural dynamics based solely on the statistical properties of such synaptic interaction matrix. This work explores the application of free random variables (FRV) to the study of large synaptic interaction matrices. Besides recovering in a straightforward way known results on eigenspectra of neural networks, we extend them to heavy-tailed distributions of interactions. More importantly, we derive analytically the behavior of eigenvector overlaps, which determine stability of the spectra. We observe that upon imposing the neuronal excitation/inhibition balance, although the eigenvalues remain unchanged, their stability dramatically decreases due to strong non-orthogonality of associated eigenvectors. It leads us to the conclusion that the understanding of the temporal evolution of asymmetric neural networks requires considering the entangled dynamics of both eigenvectors and eigenvalues, which might bear consequences for learning and memory processes in these models. Considering the success of FRV analysis in a wide variety of branches disciplines, we hope that the results presented here foster additional application of these ideas in the area of brain sciences.Recently, the two-point function associated with off-diagonal elements of the overlap matrix has become accessible within free probability [35].

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Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach

Cited by 24 publications

References 76 publications

Products of random matrices from fixed trace and induced Ginibre ensembles

Products of random matrices from fixed trace and induced Ginibre ensembles

Universal eigenvector correlations in quaternionic Ginibre ensembles

From Synaptic Interactions to Collective Dynamics in Random Neuronal Networks Models: Critical Role of Eigenvectors and Transient Behavior

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