Location of the spectrum of Kronecker random matrices

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

doi:10.1214/18-aihp894

Cited by 31 publications

(62 citation statements)

References 34 publications

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“…In particular, for Wigner‐type matrices, the global law holds under the assumptions of bounded variances and bounded moments. Our Theorems and give a moment method proof of the global law in for Wigner‐type matrices under bounded variances and Lindeberg's condition. Our new contribution is a weaker condition for the convergence of the empirical spectral distribution

μ_{n}^{M}

of M n .…”

Section: Main Results For General Wigner‐type Matricesmentioning

confidence: 99%

“…Wigner-type matrices is a special case for the Kronecker random matrices introduced in [8], and the global law has been proved in Theorem 2.7 of [8], which states the following: let H n be a Kronecker random matrix and H n be its empirical spectral distribution, then there exists a deterministic sequence of probability measure n such that H n − 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: Theorem 32 Let a N Be A General Wigner-type Matrix And W Nmentioning

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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μ_{n}^{M}

of M n .…”

Section: Main Results For General Wigner‐type Matricesmentioning

confidence: 99%

Section: Theorem 32 Let a N Be A General Wigner-type Matrix And W Nmentioning

confidence: 99%

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

Zhu

2019

Random Struct Algorithms

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“…In fact, the linearized matrix is a tensor linear combination of the independent Wigner matrices with matrix coefficients whose dimension m × m depends only on the polynomial p and is independent of N . This structure exactly corresponds to certain block matrices and more generally Kronecker random matrices introduced in [4]. We remark that the linearization technique has been widely used in the free probability community to study polynomials of random matrices on the global scale, see e.g.…”

Section: Introductionmentioning

confidence: 92%

“…Local laws for Kronecker matrix H have been studied in detail in [4] by proving concentration of its resolvent (H − zI m ⊗ I N ) −1 around the solution of corresponding matrix Dyson equation for spectral parameter z in complex upper half-plane. In contrast to the Kronecker case, to study the resolvent (P − z) −1 of our polynomial, we have to consider the generalized resolvent of the linearized matrix H, i.e., (H − zJ ⊗ I N ) −1 , where J is a rank-one m × m matrix.…”

Section: Introductionmentioning

confidence: 99%

“…After these two key obstacles cleared, we can essentially use the local law established for general Kronecker matrices in [4] under two basic conditions: (i) the solution of the underlying Dyson equation is bounded and (ii) the stability operator is invertible. These conditions are verified for homogeneous polynomials of degree two in Wigner matrices, substantially generalizing the case of anticommutator studied by Anderson in [6].…”

Section: Introductionmentioning

confidence: 99%

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Local laws for polynomials of Wigner matrices

Erdős

Krüger

Nemish

2020

Journal of Functional Analysis

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We consider general self-adjoint polynomials in several independent random matrices whose entries are centered and have the same variance. We show that under certain conditions the local law holds up to the optimal scale, i.e., the eigenvalue density on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory. We prove that these conditions hold for general homogeneous polynomials of degree two and for symmetrized products of independent matrices with i.i.d. entries, thus establishing the optimal bulk local law for these classes of ensembles. In particular, we generalize a similar result of Anderson for anticommutator. For more general polynomials our conditions are effectively checkable numerically.

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Landscape complexity beyond invariance and the elastic manifold

Ben Arous,

Bourgade,

McKenna

2023

Comm Pure Appl Math

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This paper characterizes the annealed, topological complexity (both of total critical points and of local minima) of the elastic manifold. This classical model of a disordered elastic system captures point configurations with self‐interactions in a random medium. We establish the simple versus glassy phase diagram in the model parameters, with these phases separated by a physical boundary known as the Larkin mass, confirming formulas of Fyodorov and Le Doussal. One essential, dynamical, step of the proof also applies to a general signal‐to‐noise model of soft spins in an anisotropic well, for which we prove a negative‐second‐moment threshold distinguishing positive from zero complexity. A universal near‐critical behavior appears within this phase portrait, namely quadratic near‐critical vanishing of the complexity of total critical points, and cubic near‐critical vanishing of the complexity of local minima. These two models serve as a paradigm of complexity calculations for Gaussian landscapes exhibiting few distributional symmetries, that is, beyond the invariant setting. The two main inputs for the proof are determinant asymptotics for non‐invariant random matrices from our companion paper (Ben Arous, Bourgade, McKenna 2022), and the atypical convexity and integrability of the limiting variational problems.

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Location of the spectrum of Kronecker random matrices

Cited by 31 publications

References 34 publications

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

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

Local laws for polynomials of Wigner matrices

Landscape complexity beyond invariance and the elastic manifold

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