Outlier Eigenvalues for Deformed I.I.D. Random Matrices

Bordenave, Charles; Capitaine, Mireille

doi:10.1002/cpa.21629

Cited by 40 publications

(69 citation statements)

References 70 publications

(112 reference statements)

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“…They also noted that, as σ → 0, the kernel K σ (·, ·) admits a non-trivial limit and the limiting kernel turns out to be the covariance kernel of the hyperbolic Gaussian analytic function given by (1.8). It is striking to see that for the complex Gaussian perturbation of the Jordan matrix the same limit appears in these two rather different frameworks: in [5] N → ∞ is followed by σ ↓ 0, whereas in this paper σ −1 and N are sent to infinity together with σ = N −δ for some δ > 0. However, it should also be noted that, unlike [5], here the limit is non-universal.…”

Section: Introductionmentioning

confidence: 73%

Spectrum of random perturbations of Toeplitz matrices with finite symbols

Basak

Paquette

Zeitouni

2020

Trans. Amer. Math. Soc.

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Consider an N × N Toeplitz matrix T N with symbol a(λ) := d 1 =−d 2 a λ , perturbed by an additive noise matrix N −γ E N , where the entries of E N are centered i.i.d. complex random variables of unit variance and γ > 1/2. It is known that the empirical measure of eigenvalues of the perturbed matrix converges weakly, as N → ∞, to the law of a(U ), where U is distributed uniformly on S 1 . In this paper, we consider the outliers, i.e. eigenvalues that are at a positive (N -independent) distance from a(S 1 ). We prove that there are no outliers outside spec T (a), the spectrum of the limiting Toeplitz operator, with probability approaching one, as N → ∞. In contrast, in spec T (a) \ a(S 1 ) the process of outliers converges to the point process described by the zero set of certain random analytic functions. The limiting random analytic functions can be expressed as linear combinations of the determinants of finite sub-matrices of an infinite dimensional matrix, whose entries are i.i.d. having the same law as that of E N . The coefficients in the linear combination depend on the roots of the polynomial Pz,a(λ) := (a(λ) − z)λ d 2 = 0 and semi-standard Young Tableaux with shapes determined by the number of roots of Pz,a(λ) = 0 that are greater than one in moduli. We remark that if one is interested in the case where d 1 = 0 but d 2 > 0, one may simply consider, when computing spectra, the transpose of T N or of M N = T N + ∆ N . For this reason, the restriction to d 1 > 0 does not reduce generality.

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Section: Introductionmentioning

confidence: 73%

Spectrum of random perturbations of Toeplitz matrices with finite symbols

Basak

Paquette

Zeitouni

2020

Trans. Amer. Math. Soc.

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“…However, the cavity single-instance recursions constitute an essential ingredient to arrive at the equations (21), (25) and (26) for the associated joint probability densities of the auxiliary fields of type Ω and H that characterise the typical largest eigenvalue and the statistic of the top eigenvector in the thermodynamic limit N → ∞. Moreover, the exact same equations (see (69), (111) and (70)) are found via the completely alternative replica derivation, entailing that the two methods are equivalent in the thermodynamic limit. Within the population dynamics algorithm employed to solve the stochastic recursion (21) (or equivalently (69)), we are able to identify the typical largest eigenvalue as the parameter controlling the convergence of the algorithm, and unpack the contributions coming to nodes of different degrees to the average density of the top eigenvector's components.…”

Section: Discussionmentioning

confidence: 99%

Top eigenpair statistics for weighted sparse graphs

Susca¹,

Vivo²,

Kühn³

2019

J. Phys. A: Math. Theor.

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We develop a formalism to compute the statistics of the top eigenpair of weighted sparse graphs with finite mean connectivity and bounded maximal degree. Framing the problem in terms of optimisation of a quadratic form on the sphere and introducing a fictitious temperature, we employ the cavity and replica methods to find the solution in terms of self-consistent equations for auxiliary probability density functions, which can be solved by population dynamics. This derivation allows us to identify and unpack the individual contributions to the top eigenvector's components coming from nodes of degree k. The analytical results are in perfect agreement with numerical diagonalisation of large (weighted) adjacency matrices, and are further crosschecked on the cases of random regular graphs and sparse Markov transition matrices for unbiased random walks. arXiv:1903.09852v2 [cond-mat.stat-mech] 25 Oct 2019 eigenpair problem for a differential operator [8]. The top eigenpair is also relevant in signal reconstruction problems employing algorithms based on the spectral method [9]. In the context of graph theory, the eigenvectors of both adjacency and Laplacian matrices are employed to solve combinatorial optimisation problems, such as graph 3-colouring [10] and to develop clustering and cutting techniques [11][12][13]. In particular, the top eigenvector of graphs is intimately related to the "ranking" of the nodes of the network [14]. Indeed, beyond the natural notion of ranking of a node given by its degree, the relevance of a node can be estimated from how "important" its neighbours are. The vector expressing the importance of each node is exactly the top eigenvector of the network adjacency matrix. Google PageRank algorithm works in a similar way [15,16]: the PageRanks vector is indeed the top eigenvector of a large Markov transition matrix between web pages.When the matrix J is random and symmetric with i.i.d. entries, analytical results on the statistics of the top eigenpair date back to the classical work by Füredi and Komlós [17]: the largest eigenvalue of such matrices follows a Gaussian distribution with finite variance, provided that the moments of the distribution of the entries do not scale with the matrix size. This result directly relates to the largest eigenvalue of Erdős-Rényi (E-R) [18] adjacency matrices in the case when the probability p for two nodes to be connected does not scale with the matrix size N . This result has been then extended by Janson [19] in the case when p is large. However, in our analysis we will be mostly dealing with the sparse case, i.e. when p = c/N , with c being the constant mean degree of nodes (or equivalently, the mean number of nonzero elements per row of the corresponding adjacency matrix). In this sparse regime, Krivelevich and Sudakov [20] proved a theorem stating that for any constant c the largest eigenvalue of Erdős-Rényi graph diverges slowly with N as log N/ log log N . To ensure that the largest eigenvalue remains ∼ O(1), the nodes with very large degree must be prun...

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“…We point out that the global law and the location of the spectrum for A + X, where X is an i.i.d. centered random matrix and A is a general deterministic matrix (so-called deformed ensembles), have been extensively studied, see [10,14,15,31,32]. For more references, we refer to the review [16].…”

Section: Introductionmentioning

confidence: 99%

Location of the spectrum of Kronecker random matrices

Alt¹,

Erdős²,

Krüger³

et al. 2019

Ann. Inst. H. Poincaré Probab. Statist.

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For a general class of large non-Hermitian random block matrices X we prove that there are no eigenvalues away from a deterministic set with very high probability. This set is obtained from the Dyson equation of the Hermitization of X as the self-consistent approximation of the pseudospectrum. We demonstrate that the analysis of the matrix Dyson equation from [3] offers a unified treatment of many structured matrix ensembles.

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Outlier Eigenvalues for Deformed I.I.D. Random Matrices

Cited by 40 publications

References 70 publications

Spectrum of random perturbations of Toeplitz matrices with finite symbols

Spectrum of random perturbations of Toeplitz matrices with finite symbols

Top eigenpair statistics for weighted sparse graphs

Location of the spectrum of Kronecker random matrices

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