Location of the spectrum of Kronecker random matrices

Alt, Johannes; Erdős, László; Krüger, Torben; Nemish, Yuriy

doi:10.48550/arxiv.1706.08343

Cited by 4 publications

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“…Our Theorem 3.2 and Theorem 3.4 give a moment method proof of the global law in [6] for Wigner-type matrices under bounded variances and Lindeberg's condition. In addition, our new contribution is a weaker condition for the convergence of the empirical spectral distribution µ M n of M n : the convergence of homomorphism densities t(T, W n ) for any finite trees ensure that µ M n converges weakly to a deterministic limiting measure µ.…”

Section: Main Results For General Wigner-type Matricesmentioning

confidence: 93%

“…Wigner-type matrices is a special case for the Kronecker random matrices introduced in [6], and the global law has been proved in Theorem 2.7 of [6], which states the following: let H n be a Kronecker random matrix and µ H n be its empirical spectral distribution, then there exists a deterministic sequence of probability measure µ n such that µ H n −µ n converges weakly in probability to the zero measure as n → ∞. In particular, for Wigner-type matrices, the global law holds under the assumptions of bounded variances and bounded moments.…”

Section: Main Results For General Wigner-type Matricesmentioning

confidence: 99%

“…Ding [24] used moment methods to derive the limiting spectral distribution of random block matrices for fixed number of blocks (a claim in [24] that the method extends to the growing number of blocks case is unfortunately incorrect). Recently Alt et al [6] provided a unified way to study the global law for a general class of non-Hermitian random block matrices including Wigner-type matrices.…”

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confidence: 99%

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A graphon approach to limiting spectral distributions of Wigner‐type matrices

Zhu

2019

Random Struct Algorithms

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We present a new approach, based on graphon theory, to finding the limiting spectral distributions of general Wigner‐type matrices. This approach determines the moments of the limiting measures and the equations of their Stieltjes transforms explicitly with weaker assumptions on the convergence of variance profiles than previous results. As applications, we give a new proof of the semicircle law for generalized Wigner matrices and determine the limiting spectral distributions for three sparse inhomogeneous random graph models with sparsity ω(1/n): inhomogeneous random graphs with roughly equal expected degrees, W‐random graphs and stochastic block models with a growing number of blocks. Furthermore, we show our theorems can be applied to random Gram matrices with a variance profile for which we can find the limiting spectral distributions under weaker assumptions than previous results.

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Section: Main Results For General Wigner-type Matricesmentioning

confidence: 93%

Section: Main Results For General Wigner-type Matricesmentioning

confidence: 99%

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confidence: 99%

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A graphon approach to limiting spectral distributions of Wigner‐type matrices

Zhu

2019

Random Struct Algorithms

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“…We introduce an equivalence relation on I = J × J in such a way that we first The same correlation structure is obtained if the blocks contain possibly different random matrices with independent entries (respecting only the overall Hermitian symmetry, but possibly without symmetry within each block), see e.g. the ensemble discussed in [5]. Furthermore, one may combine the block matrix model with a polynomially decaying model from Example 2.10 to construct yet another example for which Theorem 2.2 is applicable.…”

Section: ) Independent Of the Chosen Neighbourhood Systems As For Ass...mentioning

confidence: 99%

Random Matrices With Slow Correlation Decay

Erdős

Krüger

Schröder

2019

Forum of Mathematics, Sigma

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203

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We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the recent result of [3] to allow slow correlation decay and arbitrary expectation. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion. ContentsRANDOM MATRICES WITH SLOW CORRELATION DECAY 2 of the individual eigenvalues follows a universal distribution, independent of the specifics of the random matrix itself. The former is commonly called a local law, whereas the latter is known as the Wigner-Dyson-Mehta (WDM) universality conjecture, first envisioned by Wigner in the 1950's and formalized later by Dyson and Mehta in the 1960's [36]. In fact, the conjecture extends beyond the customary random matrix ensembles in probability theory and is believed to hold for any random operator in the delocalization regime of the Anderson metal-insulator phase transition. Given this profound universality conjecture for general disordered quantum systems, the ultimate goal of local spectral analysis of large random matrices is to prove the WDM conjecture for the largest possible class of matrix ensembles. In the current paper we complete this program for random matrices with a general, slow correlation decay among its matrix elements. Previous works covered only correlations with such a fast decay that, in a certain sense, they could be treated as a perturbation of the independent model. Here we present a new method that goes well beyond the perturbative regime. It relies on a novel multi-scale version of the cumulant expansion and its rigorous Feynman diagrammatic representation that can be useful for other problems as well. To put our work in context, we now explain the previous results.In the last ten years a powerful new approach, the three-step strategy has been developed to resolve WDM universality problems, see [19] for a summary. In particular, the WDM conjecture in its classical form, stated for Wigner matrices with a general distribution of the entries, has been proven with this strategy in [14,15,21]; an independent proof for the Hermitian symmetry class was given in [42]. Recent advances have crystallized that the only model dependent step in this strategy is the first one, the local law. The other two steps, the fast relaxation to equilibrium of the Dyson Brownian motion and the approximation by Gaussian divisible ensembles, have been formulated as very general "black-box" tools whose only input is the local law [17,31,32]. Thus the proof of the WDM universality, at least for mean field ensembles, is automatically reduced to obtaining a local law.Both local law and universality have first been established for Wigner matrices, which are real symmetric or complex Hermitian N × N matrices with mean-zero entries which are independent and identically distributed (i.i.d.) up to symmetry [15,16]. For Wigner matrices it has long been known that the l...

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“…Away from supp ν, the convergence rate in (2.8a) and (2.8b) can be improved and thus the condition Im ζ ≥ p −1+γ can be removed. See [5] for Gram matrices and [4] for Kronecker matrices.…”

Section: 8b)mentioning

confidence: 99%

Singularities of the density of states of random Gram matrices

Alt¹

2017

Electron. Commun. Probab.

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For large random matrices X with independent, centered entries but not necessarily identical variances, the eigenvalue density of XX * is well-approximated by a deterministic measure on R. We show that the density of this measure has only square and cubic-root singularities away from zero. We also extend the bulk local law in [5] to the vicinity of these singularities. Main results Structure of the solution to the Dyson equationLet (X 1 , S 1 , π 1 ) and (X 2 , S 2 , π 2 ) be two finite measure spaces such that π 1 (X 1 ) and π 2 (X 2 ) are strictly positive. Moreover, we denote the spaces of bounded and measurable functions on X 1 and X 2 by

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Location of the spectrum of Kronecker random matrices

Cited by 4 publications

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A graphon approach to limiting spectral distributions of Wigner‐type matrices

A graphon approach to limiting spectral distributions of Wigner‐type matrices

Random Matrices With Slow Correlation Decay

Singularities of the density of states of random Gram matrices

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