scholarly journals Edge universality for deformed Wigner matrices

2015 ◽  
Vol 27 (08) ◽  
pp. 1550018 ◽  
Author(s):  
Ji Oon Lee ◽  
Kevin Schnelli
Keyword(s):  
Large Class ◽  
Random Matrices ◽  
Diagonal Matrix ◽  
Wigner Matrix ◽  
Wigner Matrices ◽  
Large N ◽  
The Matrix

We consider N × N random matrices of the form H = W + V where W is a real symmetric Wigner matrix and V a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W and we choose V so that the eigenvalues of W and V are typically of the same order. For a large class of diagonal matrices V, we show that the rescaled distribution of the extremal eigenvalues is given by the Tracy–Widom distribution F1 in the limit of large N. Our proofs also apply to the complex Hermitian setting, i.e. when W is a complex Hermitian Wigner matrix.

scholarly journals Gaussian fluctuations for linear spectral statistics of deformed Wigner matrices

2019 ◽  
Vol 09 (03) ◽  
pp. 2050011
Author(s):  
Hong Chang Ji ◽  
Ji Oon Lee
Keyword(s):  
Large Class ◽  
Random Matrices ◽  
Gaussian Distribution ◽  
Diagonal Matrix ◽  
Limiting Distributions ◽  
Test Function ◽  
Wigner Matrix ◽  
Wigner Matrices ◽  
Gaussian Fluctuations ◽  
Spectral Statistics

We consider large-dimensional Hermitian or symmetric random matrices of the form [Formula: see text], where [Formula: see text] is a Wigner matrix and [Formula: see text] is a real diagonal matrix whose entries are independent of [Formula: see text]. For a large class of diagonal matrices [Formula: see text], we prove that the fluctuations of linear spectral statistics of [Formula: see text] for [Formula: see text] test function can be decomposed into that of [Formula: see text] and of [Formula: see text], and that each of those weakly converges to a Gaussian distribution. We also calculate the formulae for the means and variances of the limiting distributions.

scholarly journals ON FINITE RANK DEFORMATIONS OF WIGNER MATRICES II: DELOCALIZED PERTURBATIONS

2013 ◽  
Vol 02 (01) ◽  
pp. 1250015 ◽  
Author(s):  
DAVID RENFREW ◽  
ALEXANDER SOSHNIKOV
Keyword(s):  
Random Matrices ◽  
Finite Rank ◽  
Fourth Moment ◽  
Wigner Matrices ◽  
Wigner Random Matrices ◽  
The Matrix

We study the distribution of the outliers in the spectrum of finite rank deformations of Wigner random matrices. We assume that the matrix entries have finite fourth moment and extend the results by Capitaine, Donati-Martin, and Féral for perturbations whose eigenvectors are delocalized.

scholarly journals Nonuniversality of fluctuations of outliers for Hermitian polynomials in a complex Wigner matrix and a spiked diagonal matrix

2019 ◽  
Vol 09 (04) ◽  
pp. 2050013
Author(s):  
Mireille Capitaine
Keyword(s):  
Probability Theory ◽  
Free Probability ◽  
Diagonal Matrix ◽  
Free Probability Theory ◽  
Wigner Matrix ◽  
Wigner Matrices ◽  
Hermitian Polynomial ◽  
Hermitian Polynomials

We study the fluctuations associated to the a.s. convergence of the outliers established by Belinschi–Bercovici–Capitaine of an Hermitian polynomial in a complex Wigner matrix and a spiked deterministic real diagonal matrix. Thus, we extend the nonuniversality phenomenon established by Capitaine–Donati-Martin–Féral for additive deformations of complex Wigner matrices, to any Hermitian polynomial. The result is described using the operator-valued subordination functions of free probability theory.

scholarly journals LSV-Based Tail Inequalities for Sums of Random Matrices

2017 ◽  
Vol 29 (1) ◽  
pp. 247-262 ◽  
Author(s):  
Chao Zhang ◽  
Lei Du ◽  
Dacheng Tao
Keyword(s):  
Machine Learning ◽  
Laplace Transform ◽  
Random Matrices ◽  
Singular Value ◽  
Diagonal Matrix ◽  
New Method ◽  
Learning Models ◽  
The Matrix ◽  
Diagonalization Method ◽  
Machine Learning Models

The techniques of random matrices have played an important role in many machine learning models. In this letter, we present a new method to study the tail inequalities for sums of random matrices. Different from other work (Ahlswede & Winter, 2002 ; Tropp, 2012 ; Hsu, Kakade, & Zhang, 2012 ), our tail results are based on the largest singular value (LSV) and independent of the matrix dimension. Since the LSV operation and the expectation are noncommutative, we introduce a diagonalization method to convert the LSV operation into the trace operation of an infinitely dimensional diagonal matrix. In this way, we obtain another version of Laplace-transform bounds and then achieve the LSV-based tail inequalities for sums of random matrices.

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

scholarly journals Effective Action and Measure in Matrix Model of IIB Superstrings

1997 ◽  
Vol 12 (31) ◽  
pp. 2331-2340 ◽  
Author(s):  
L. Chekhov ◽  
K. Zarembo
Keyword(s):  
Matrix Model ◽  
Effective Action ◽  
Continuum Limit ◽  
Auxiliary Field ◽  
Large N ◽  
The Matrix ◽  
Reparametrization Invariance ◽  
The Continuum ◽  
Induced Measure

We calculate an effective action and measure induced by the integration over the auxiliary field in the matrix model recently proposed to describe IIB superstrings. It is shown that the measure of integration over the auxiliary matrix is uniquely determined by locality and reparametrization invariance of the resulting effective action. The large-N limit of the induced measure for string coordinates is discussed in detail. It is found to be ultralocal and, thus, is possibly irrelevant in the continuum limit. The model of the GKM type is considered in relation to the effective action problem.

scholarly journals Higher-order Darboux transformations for two-dimensional Dirac systems with diagonal matrix potential

2021 ◽  
Vol 2090 (1) ◽  
pp. 012038
Author(s):  
A Schulze-Halberg
Keyword(s):  
Dirac Equation ◽  
Explicit Form ◽  
Higher Order ◽  
Diagonal Matrix ◽  
Two Dimensional ◽  
Darboux Transformations ◽  
Dirac Systems ◽  
Matrix Potential ◽  
De Vries ◽  
The Matrix

Abstract We construct the explicit form of higher-order Darboux transformations for the two-dimensional Dirac equation with diagonal matrix potential. The matrix potential entries can depend arbitrarily on the two variables. Our construction is based on results for coupled Korteweg-de Vries equations [27].

scholarly journals An asymptotic property of large matrices with identically distributed Boolean independent entries

2019 ◽  
Vol 22 (04) ◽  
pp. 1950024
Author(s):  
Mihai Popa ◽  
Zhiwei Hao
Keyword(s):  
Recent Result ◽  
Recent Work ◽  
Random Matrices ◽  
Asymptotic Property ◽  
Limit Distribution ◽  
Asymptotic Independence ◽  
Moment Conditions ◽  
The Matrix ◽  
Combinatorial Condition ◽  
Independence Relations

Motivated by the recent work on asymptotic independence relations for random matrices with non-commutative entries, we investigate the limit distribution and independence relations for large matrices with identically distributed and Boolean independent entries. More precisely, we show that, under some moment conditions, such random matrices are asymptotically [Formula: see text]-diagonal and Boolean independent from each other. This paper also gives a combinatorial condition under which such matrices are asymptotically Boolean independent from the matrix obtained by permuting the entries (thus extending a recent result in Boolean probability). In particular, we show that the random matrices considered are asymptotically Boolean independent from some of their partial transposes. The main results of the paper are based on combinatorial techniques.

From random matrices to random tensors

2021 ◽  
pp. 166-177
Author(s):  
Adrian Tanasa
Keyword(s):  
Matrix Models ◽  
Random Matrices ◽  
Mathematical Physics ◽  
Natural Generalization ◽  
Random Surface ◽  
The Third ◽  
Planar Surfaces ◽  
Tensor Models ◽  
Feynman Graphs ◽  
The Matrix

After a brief presentation of random matrices as a random surface QFT approach to 2D quantum gravity, we focus on two crucial mathematical physics results: the implementation of the large N limit (N being here the size of the matrix) and of the double-scaling mechanism for matrix models. It is worth emphasizing that, in the large N limit, it is the planar surfaces which dominate. In the third section of the chapter we introduce tensor models, seen as a natural generalization, in dimension higher than two, of matrix models. The last section of the chapter presents a potential generalisation of the Bollobás–Riordan polynomial for tensor graphs (which are the Feynman graphs of the perturbative expansion of QFT tensor models).

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