363 results on '"Random matrices"'
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2. Sums of GUE matrices and concentration of hives from correlation decay of eigengaps.
- Author
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Narayanan, Hariharan, Sheffield, Scott, and Tao, Terence
- Subjects
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MATRICES (Mathematics) , *RANDOM matrices , *EIGENVALUES , *LOGICAL prediction - Abstract
Associated to two given sequences of eigenvalues λ 1 ≥ ⋯ ≥ λ n and μ 1 ≥ ⋯ ≥ μ n is a natural polytope, the polytope of augmented hives with the specified boundary data, which is associated to sums of random Hermitian matrices with these eigenvalues. As a first step towards the asymptotic analysis of random hives, we show that if the eigenvalues are drawn from the GUE ensemble, then the associated augmented hives exhibit concentration as n → ∞ . Our main ingredients include a representation due to Speyer of augmented hives involving a supremum of linear functions applied to a product of Gelfand–Tsetlin polytopes; known results by Klartag on the KLS conjecture in order to handle the aforementioned supremum; covariance bounds of Cipolloni–Erdős–Schröder of eigenvalue gaps of GUE; and the use of the theory of determinantal processes to analyze the GUE minor process. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. Joint distribution of the cokernels of random p-adic matrices II.
- Author
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Jung, Jiwan and Lee, Jungin
- Subjects
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MATRIX rings , *RINGS of integers , *RANDOM matrices , *INTEGERS , *P-adic analysis - Abstract
In this paper, we study the combinatorial relations between the cokernels cok (A n + p x i I n ) ( 1 ≤ i ≤ m ), where A n is an n × n matrix over the ring of p-adic integers ℤ p , I n is the n × n identity matrix and x 1 , ... , x m are elements of ℤ p whose reductions modulo p are distinct. For a positive integer m ≤ 4 and given x 1 , ... , x m ∈ ℤ p , we determine the set of m-tuples of finitely generated ℤ p -modules (H 1 , ... , H m) for which (cok (A n + p x 1 I n ) , ... , cok (A n + p x m I n )) = (H 1 , ... , H m) for some matrix A n . We also prove that if A n is an n × n Haar random matrix over ℤ p for each positive integer n, then the joint distribution of cok (A n + p x i I n ) ( 1 ≤ i ≤ m ) converges as n → ∞ . [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
4. Infinite-dimensional stochastic differential equations arising from Airy random point fields
- Author
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Osada, Hirofumi and Tanemura, Hideki
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- 2024
- Full Text
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5. Universality and Sharp Matrix Concentration Inequalities
- Author
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Brailovskaya, Tatiana and van Handel, Ramon
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- 2024
- Full Text
- View/download PDF
6. A Stirling-Type Formula for the Distribution of the Length of Longest Increasing Subsequences.
- Author
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Bornemann, Folkmar
- Subjects
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RANDOM matrices , *DISTRIBUTION (Probability theory) - Abstract
The discrete distribution of the length of longest increasing subsequences in random permutations of n integers is deeply related to random matrix theory. In a seminal work, Baik, Deift and Johansson provided an asymptotics in terms of the distribution of the scaled largest level of the large matrix limit of GUE. As a numerical approximation, however, this asymptotics is inaccurate for small n and has a slow convergence rate, conjectured to be just of order n - 1 / 3 . Here, we suggest a different type of approximation, based on Hayman's generalization of Stirling's formula. Such a formula gives already a couple of correct digits of the length distribution for n as small as 20 but allows numerical evaluations, with a uniform error of apparent order n - 2 / 3 , for n as large as 10 12 , thus closing the gap between a table of exact values (compiled for up to n = 1000 ) and the random matrix limit. Being much more efficient and accurate than Monte Carlo simulations, the Stirling-type formula allows for a precise numerical understanding of the first few finite size correction terms to the random matrix limit. From this we derive expansions of the expected value and variance of the length, exhibiting several more terms than previously put forward. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
7. Mesoscopic central limit theorem for non-Hermitian random matrices.
- Author
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Cipolloni, Giorgio, Erdős, László, and Schröder, Dominik
- Subjects
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CENTRAL limit theorem , *RANDOM matrices , *COMPLEX matrices , *MATHEMATICS , *NONEXPANSIVE mappings - Abstract
We prove that the mesoscopic linear statistics ∑ i f (n a (σ i - z 0)) of the eigenvalues { σ i } i of large n × n non-Hermitian random matrices with complex centred i.i.d. entries are asymptotically Gaussian for any H 0 2 -functions f around any point z 0 in the bulk of the spectrum on any mesoscopic scale 0 < a < 1 / 2 . This extends our previous result (Cipolloni et al. in Commun Pure Appl Math, 2019. arXiv:1912.04100), that was valid on the macroscopic scale, a = 0 , to cover the entire mesoscopic regime. The main novelty is a local law for the product of resolvents for the Hermitization of X at spectral parameters z 1 , z 2 with an improved error term in the entire mesoscopic regime | z 1 - z 2 | ≫ n - 1 / 2 . The proof is dynamical; it relies on a recursive tandem of the characteristic flow method and the Green function comparison idea combined with a separation of the unstable mode of the underlying stability operator. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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8. Single eigenvalue fluctuations of general Wigner-type matrices.
- Author
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Landon, Benjamin, Lopatto, Patrick, and Sosoe, Philippe
- Subjects
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RANDOM matrices , *CENTRAL limit theorem , *EIGENVALUES , *MATRICES (Mathematics) , *DENSITY of states - Abstract
We consider the single eigenvalue fluctuations of random matrices of general Wigner-type, under a one-cut assumption on the density of states. For eigenvalues in the bulk, we prove that the asymptotic fluctuations of a single eigenvalue around its classical location are Gaussian with a universal variance. Our method is based on a dynamical approach to mesoscopic linear spectral statistics which reduces their behavior on short scales to that on larger scales. We prove a central limit theorem for linear spectral statistics on larger scales via resolvent techniques and show that for certain classes of test functions, the leading-order contribution to the variance agrees with the GOE/GUE cases. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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9. Extremal singular values of random matrix products and Brownian motion on GL(N,C).
- Author
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Ahn, Andrew
- Subjects
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MATRIX multiplications , *RANDOM matrices , *BROWNIAN motion , *VALUES (Ethics) , *GENERATING functions , *BESSEL functions - Abstract
We establish universality for the largest singular values of products of random matrices with right unitarily invariant distributions, in a regime where the number of matrix factors and size of the matrices tend to infinity simultaneously. The behavior of the largest log singular values coincides with the large N limit of Dyson Brownian motion with a characteristic drift vector consisting of equally spaced coordinates, which matches the large N limit of the largest log singular values of Brownian motion on GL (N , C) . Our method utilizes the formalism of multivariate Bessel generating functions, also known as spherical transforms, to obtain and analyze combinatorial expressions for observables of these processes. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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10. Fluctuation Moments Induced by Conjugation with Asymptotically Liberating Random Matrix Ensembles.
- Author
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Vazquez-Becerra, Josue
- Abstract
Independent Haar-unitary random matrices and independent Haar-orthogonal random matrices are known to be asymptotically liberating ensembles, and they give rise to asymptotic free independence when used for conjugation of constant matrices. G. Anderson and B. Farrel showed that a certain family of discrete random unitary matrices can actually be used to the same end. In this paper, we investigate fluctuation moments and higher-order moments induced on constant matrices by conjugation with asymptotically liberating ensembles. We show for the first time that the fluctuation moments associated with second-order free independence can be obtained from conjugation with an ensemble consisting of signed permutation matrices and the discrete Fourier transform matrix. We also determine fluctuation moments induced by various related ensembles where we do not get known expressions but others related to traffic free independence. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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11. Matrix Whittaker processes.
- Author
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Arista, Jonas, Bisi, Elia, and O'Connell, Neil
- Subjects
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RANDOM matrices , *PARTITION functions , *WHITTAKER functions , *WISHART matrices , *MATRIX functions , *RANDOM graphs , *DIRECTED graphs - Abstract
We study a discrete-time Markov process on triangular arrays of matrices of size d ≥ 1 , driven by inverse Wishart random matrices. The components of the right edge evolve as multiplicative random walks on positive definite matrices with one-sided interactions and can be viewed as a d-dimensional generalisation of log-gamma polymer partition functions. We establish intertwining relations to prove that, for suitable initial configurations of the triangular process, the bottom edge has an autonomous Markovian evolution with an explicit transition kernel. We then show that, for a special singular initial configuration, the fixed-time law of the bottom edge is a matrix Whittaker measure, which we define. To achieve this, we perform a Laplace approximation that requires solving a constrained minimisation problem for certain energy functions of matrix arguments on directed graphs. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
12. Detection Thresholds in Very Sparse Matrix Completion.
- Author
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Bordenave, Charles, Coste, Simon, and Nadakuditi, Raj Rao
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SPARSE matrices , *RANDOM matrices , *SINGULAR value decomposition , *PRINCIPAL components analysis , *EIGENVECTORS , *EIGENVALUES - Abstract
We study the matrix completion problem: an underlying m × n matrix P is low rank, with incoherent singular vectors, and a random m × n matrix A is equal to P on a (uniformly) random subset of entries of size dn. All other entries of A are equal to zero. The goal is to retrieve information on P from the observation of A. Let A 1 be the random matrix where each entry of A is multiplied by an independent { 0 , 1 } -Bernoulli random variable with parameter 1/2. This paper is about when, how and why the non-Hermitian eigen-spectra of the matrices A 1 (A - A 1) ∗ and (A - A 1) ∗ A 1 captures more of the relevant information about the principal component structure of A than the eigen-spectra of A A ∗ and A ∗ A . We show that the eigenvalues of the asymmetric matrices A 1 (A - A 1) ∗ and (A - A 1) ∗ A 1 with modulus greater than a detection threshold are asymptotically equal to the eigenvalues of P P ∗ and P ∗ P and that the associated eigenvectors are aligned as well. The central surprise is that by intentionally inducing asymmetry and additional randomness via the A 1 matrix, we can extract more information than if we had worked with the singular value decomposition (SVD) of A. The associated detection threshold is asymptotically exact and is non-universal since it explicitly depends on the element-wise distribution of the underlying matrix P. We show that reliable, statistically optimal but not perfect matrix recovery, via a universal data-driven algorithm, is possible above this detection threshold using the information extracted from the asymmetric eigen-decompositions. Averaging the left and right eigenvectors provably improves estimation accuracy but not the detection threshold. Our results encompass the very sparse regime where d is of order 1 where matrix completion via the SVD of A fails or produces unreliable recovery. We define another variant of this asymmetric principal component analysis procedure that bypasses the randomization step and has a detection threshold that is smaller by a constant factor but with a computational cost that is larger by a polynomial factor of the number of observed entries. Both detection thresholds allow to go beyond the barrier due to the well-known information theoretical limit d ≍ log n for exact matrix completion found in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
13. Matrix concentration inequalities and free probability.
- Author
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Bandeira, Afonso S., Boedihardjo, March T., and van Handel, Ramon
- Subjects
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RANDOM matrices , *NONCOMMUTATIVE algebras , *PROBABILITY theory , *RANDOM graphs , *MATRIX inequalities - Abstract
A central tool in the study of nonhomogeneous random matrices, the noncommutative Khintchine inequality, yields a nonasymptotic bound on the spectral norm of general Gaussian random matrices X = ∑ i g i A i where g i are independent standard Gaussian variables and A i are matrix coefficients. This bound exhibits a logarithmic dependence on dimension that is sharp when the matrices A i commute, but often proves to be suboptimal in the presence of noncommutativity. In this paper, we develop nonasymptotic bounds on the spectrum of arbitrary Gaussian random matrices that can capture noncommutativity. These bounds quantify the degree to which the spectrum of X is captured by that of a noncommutative model X free that arises from free probability theory. This "intrinsic freeness" phenomenon provides a powerful tool for the study of various questions that are outside the reach of classical methods of random matrix theory. Our nonasymptotic bounds are easily applicable in concrete situations, and yield sharp results in examples where the noncommutative Khintchine inequality is suboptimal. When combined with a linearization argument, our bounds imply strong asymptotic freeness for a remarkably general class of Gaussian random matrix models that may be very sparse, have dependent entries, and lack any special symmetries. When combined with a universality principle, our bounds extend beyond the Gaussian setting to general sums of independent random matrices. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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14. Random walks on SL2(C): spectral gap and limit theorems.
- Author
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Dinh, Tien-Cuong, Kaufmann, Lucas, and Wu, Hao
- Subjects
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LIMIT theorems , *RANDOM walks , *MARKOV operators , *SOBOLEV spaces , *FUNCTION spaces , *MATRIX norms - Abstract
We obtain various new limit theorems for random walks on SL 2 (C) under low moment conditions. For non-elementary measures with a finite second moment we prove a Local Limit Theorem for the norm cocycle, yielding the optimal version of a theorem of É. Le Page. For measures with a finite third moment, we obtain the Local Limit Theorem for the matrix coefficients, improving a recent result of Grama-Quint-Xiao and the authors, and Berry–Esseen bounds with optimal rate O (1 / n) for the norm cocycle and the matrix coefficients. The main tool is a detailed study of the spectral properties of the Markov operator and its purely imaginary perturbations acting on different function spaces. We introduce, in particular, a new function space derived from the Sobolev space W 1 , 2 that provides uniform estimates. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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15. Joint distribution of the cokernels of random p-adic matrices.
- Author
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Lee, Jungin
- Subjects
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RANDOM matrices , *HAAR integral , *POLYNOMIALS - Abstract
In this paper, we study the joint distribution of the cokernels of random p-adic matrices. Let p be a prime and let P 1 (t) , ... , P l (t) ∈ ℤ p [ t ] be monic polynomials whose reductions modulo p in 픽 p [ t ] are distinct and irreducible. We determine the limit of the joint distribution of the cokernels cok (P 1 (A)) , ... , cok (P l (A)) for a random n × n matrix A over ℤ p with respect to the Haar measure as n → ∞ . By applying the linearization of a random matrix model, we also provide a conjecture which generalizes this result. Finally, we provide a sufficient condition that the cokernels cok (A) and cok (A + B n) become independent as n → ∞ , where B n is a fixed n × n matrix over ℤ p for each n and A is a random n × n matrix over ℤ p . [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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16. Dyson’s Model in Infinite Dimensions Is Irreducible
- Author
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Osada, Hirofumi, Tsuboi, Ryosuke, Chen, Zhen-Qing, editor, Takeda, Masayoshi, editor, and Uemura, Toshihiro, editor
- Published
- 2022
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17. Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics.
- Author
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Kieburg, Mario, Li, Shi-Hao, Zhang, Jiyuan, and Forrester, Peter J.
- Subjects
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RANDOM matrices , *PROBABILITY density function , *UNITARY groups , *STATISTICS , *BROWNIAN motion , *KERNEL functions , *HYPERGEOMETRIC series , *FISHER exact test , *EIGENVALUES - Abstract
A framework to study the eigenvalue probability density function for products of unitary random matrices with an invariance property is developed. This involves isolating a class of invariant unitary matrices, to be referred to as cyclic Pólya ensembles, and examining their properties with respect to the spherical transform on U (N) . Included in the cyclic Pólya ensemble class are Haar invariant unitary matrices, the circular Jacobi ensemble, known in relation to the Fisher-Hartwig singularity in the theory of Toeplitz determinants, as well as the heat kernel for Brownian motion on the unitary group. We define cyclic Pólya frequency functions and show their relation to the cyclic Pólya ensembles, and give a uniqueness statement for the corresponding weights. The natural appearance of bilateral hypergeometric series is highlighted, with this special function playing the role of the Meijer G-function in the transform theory of unitary invariant product of positive definite matrices. We construct a family of functions forming bi-orthonormal pairs which underly the correlation kernel of the corresponding determinantal point processes, and furthermore obtain an integral formula for the correlation kernel involving just two of these functions. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
18. A CLT for the characteristic polynomial of random Jacobi matrices, and the GβE.
- Author
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Augeri, Fanny, Butez, Raphael, and Zeitouni, Ofer
- Subjects
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JACOBI operators , *JACOBI polynomials , *RANDOM matrices , *CENTRAL limit theorem , *LIMIT theorems - Abstract
We prove a central limit theorem for the real part of the logarithm of the characteristic polynomial of random Jacobi matrices. Our results cover the G β E models for β > 0 . [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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19. Free Component Analysis: Theory, Algorithms and Applications.
- Author
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Nadakuditi, Raj Rao and Wu, Hao
- Subjects
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INDEPENDENT component analysis , *RANDOM matrices , *PROBABILITY theory , *BLIND source separation , *ALGORITHMS - Abstract
We describe a method for unmixing mixtures of freely independent random variables in a manner analogous to the independent component analysis (ICA)-based method for unmixing independent random variables from their additive mixtures. Random matrices play the role of free random variables in this context so the method we develop, which we call free component analysis (FCA), unmixes matrices from additive mixtures of matrices. Thus, while the mixing model is standard, the novelty and difference in unmixing performance comes from the introduction of a new statistical criteria, derived from free probability theory, that quantify freeness analogous to how kurtosis and entropy quantify independence. We describe the theory, the various algorithms, and compare FCA to vanilla ICA which does not account for spatial or temporal structure. We highlight why the statistical criteria make FCA also vanilla despite its matricial underpinnings and show that FCA performs comparably to, and sometimes better than, (vanilla) ICA in every application, such as image and speech unmixing, where ICA has been known to succeed. Our computational experiments suggest that not-so-random matrices, such as images and short-time Fourier transform matrix of waveforms are (closer to being) freer "in the wild" than we might have theoretically expected. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
20. Quenched universality for deformed Wigner matrices.
- Author
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Cipolloni, Giorgio, Erdős, László, and Schröder, Dominik
- Subjects
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MATRICES (Mathematics) , *REAL variables , *RANDOM matrices , *RANDOM variables , *EIGENVALUES , *BROWNIAN motion - Abstract
Following E. Wigner's original vision, we prove that sampling the eigenvalue gaps within the bulk spectrum of a fixed (deformed) Wigner matrix H yields the celebrated Wigner-Dyson-Mehta universal statistics with high probability. Similarly, we prove universality for a monoparametric family of deformed Wigner matrices H + x A with a deterministic Hermitian matrix A and a fixed Wigner matrix H, just using the randomness of a single scalar real random variable x. Both results constitute quenched versions of bulk universality that has so far only been proven in annealed sense with respect to the probability space of the matrix ensemble. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
21. Halting Time is Predictable for Large Models: A Universality Property and Average-Case Analysis.
- Author
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Paquette, Courtney, van Merriënboer, Bart, Paquette, Elliot, and Pedregosa, Fabian
- Subjects
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DISTRIBUTION (Probability theory) , *RANDOM matrices , *COMPUTER simulation - Abstract
Average-case analysis computes the complexity of an algorithm averaged over all possible inputs. Compared to worst-case analysis, it is more representative of the typical behavior of an algorithm, but remains largely unexplored in optimization. One difficulty is that the analysis can depend on the probability distribution of the inputs to the model. However, we show that this is not the case for a class of large-scale problems trained with first-order methods including random least squares and one-hidden layer neural networks with random weights. In fact, the halting time exhibits a universality property: it is independent of the probability distribution. With this barrier for average-case analysis removed, we provide the first explicit average-case convergence rates showing a tighter complexity not captured by traditional worst-case analysis. Finally, numerical simulations suggest this universality property holds for a more general class of algorithms and problems. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
22. Wavelet eigenvalue regression in high dimensions.
- Author
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Abry, Patrice, Boniece, B. Cooper, Didier, Gustavo, and Wendt, Herwig
- Abstract
In this paper, we construct the wavelet eigenvalue regression methodology (Abry and Didier in J Multivar Anal 168:75–104, 2018a; in Bernoulli 24(2):895–928, 2018b) in high dimensions. We assume that possibly non-Gaussian, finite-variance p-variate measurements are made of a low-dimensional r-variate ( r ≪ p ) fractional stochastic process with non-canonical scaling coordinates and in the presence of additive high-dimensional noise. The measurements are correlated both time-wise and between rows. Building upon the asymptotic and large scale properties of wavelet random matrices in high dimensions, the wavelet eigenvalue regression is shown to be consistent and, under additional assumptions, asymptotically Gaussian in the estimation of the fractal structure of the system. We further construct a consistent estimator of the effective dimension r of the system that significantly increases the robustness of the methodology. The estimation performance over finite samples is studied by means of simulations. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
23. Deformed Fréchet law for Wigner and sample covariance matrices with tail in crossover regime.
- Author
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Han, Yi
- Subjects
- *
DISTRIBUTION (Probability theory) , *SPARSE matrices , *COVARIANCE matrices , *RANDOM matrices , *SYMMETRIC matrices , *REGULAR graphs - Abstract
Given An:=1n(aij)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A_n:=\frac{1}{\sqrt{n}}(a_{ij})$$\end{document} an n×n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n\times n$$\end{document} symmetric random matrix, with elements above the diagonal given by i.i.d. random variables having mean zero and unit variance. It is known that when limx→∞x4P(|aij|>x)=0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lim _{x\rightarrow \infty }x^4\mathbb {P}(|a_{ij}|>x)=0$$\end{document}, then fluctuation of the largest eigenvalue of An\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A_n$$\end{document} follows a Tracy–Widom distribution. When the law of aij\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a_{ij}$$\end{document} is regularly varying with index α∈(0,4)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \in (0,4)$$\end{document}, then the largest eigenvalue has a Fréchet distribution. An intermediate regime is recently uncovered in Diaconu (Ann Probab 51(2):774–804, 2023): when limx→∞x4P(|aij|>x)=c∈(0,∞)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lim _{x\rightarrow \infty }x^4\mathbb {P}(|a_{ij}|>x)=c\in (0,\infty )$$\end{document}, then the law of the largest eigenvalue converges to a deformed Fréchet distribution. In this work we vastly extend the scope where the latter distribution may arise. We show that the same deformed Fréchet distribution arises (1) for sparse Wigner matrices with an average of nΩ(1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n^{\Omega }(1)$$\end{document} nonzero entries on each row; (2) for periodically banded Wigner matrices with bandwidth pn=nO(1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p_n=n^{O(1)}$$\end{document}; and more generally for weighted adjacency matrices of any kn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$k_n$$\end{document}-regular graphs with kn=nΩ(1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$k_n=n^{\Omega (1)}$$\end{document}. In all these cases, we further prove that the joint distribution of the finitely many largest eigenvalues of An\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A_n$$\end{document} converge to a deformed Poisson process, and that eigenvectors of the outlying eigenvalues of An\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A_n$$\end{document} are localized, implying a mobility edge phenomenon at the spectral edge 2 for Wigner matrices. The sparser case with average degree no(1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n^{o(1)}$$\end{document} is also explored. Our technique extends to sample covariance matrices, proving for the first time that its largest eigenvalue still follows a deformed Fréchet distribution, assuming the matrix entries satisfy limx→∞x4P(|aij|>x)=c∈(0,∞)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lim _{x\rightarrow \infty }x^4\mathbb {P}(|a_{ij}|>x)=c\in (0,\infty )$$\end{document}. The proof utilizes a universality result recently established by Brailovskaya and Van Handel (Universality and sharp matrix concentration inequalities, 2022). [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
24. Empirical Spectral Distributions of Sparse Random Graphs
- Author
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Dembo, Amir, Lubetzky, Eyal, Zhang, Yumeng, Dereich, Steffen, Series Editor, Khoshnevisan, Davar, Series Editor, Kyprianou, Andreas E., Series Editor, Resnick, Sidney I., Series Editor, Vares, Maria Eulália, editor, Fernández, Roberto, editor, Fontes, Luiz Renato, editor, and Newman, Charles M., editor
- Published
- 2021
- Full Text
- View/download PDF
25. Statistical properties of simple random-effects models for genetic heritability
- Author
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Steinsaltz, David, Dahl, Andrew, and Wachter, Kenneth W
- Subjects
Mathematical Sciences ,Statistics ,Genetics ,2.5 Research design and methodologies (aetiology) ,Aetiology ,Heritability ,random-effects models ,random matrices ,Marcenko-Pastur distribution ,GCTA ,60B20 ,62F10 ,Marcenko–Pastur distribution ,Primary 92D10 ,heritability ,secondary 62P10 - Abstract
Random-effects models are a popular tool for analysing total narrow-sense heritability for quantitative phenotypes, on the basis of large-scale SNP data. Recently, there have been disputes over the validity of conclusions that may be drawn from such analysis. We derive some of the fundamental statistical properties of heritability estimates arising from these models, showing that the bias will generally be small. We show that that the score function may be manipulated into a form that facilitates intelligible interpretations of the results. We go on to use this score function to explore the behavior of the model when certain key assumptions of the model are not satisfied - shared environment, measurement error, and genetic effects that are confined to a small subset of sites. The variance and bias depend crucially on the variance of certain functionals of the singular values of the genotype matrix. A useful baseline is the singular value distribution associated with genotypes that are completely independent - that is, with no linkage and no relatedness - for a given number of individuals and sites. We calculate the corresponding variance and bias for this setting.
- Published
- 2018
26. Gaussian unitary ensemble in random lozenge tilings.
- Author
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Aggarwal, Amol and Gorin, Vadim
- Subjects
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RANDOM matrices , *STOCHASTIC processes - Abstract
This paper establishes a universality result for scaling limits of uniformly random lozenge tilings of large domains. We prove that whenever the boundary of the domain has three adjacent straight segments inclined under 120 degrees (measured in the direction internal to the domain) to each other, the asymptotics of tilings near the middle segment is described by the GUE–corners process of random matrix theory. An important step in our argument is to show that fluctuations of the height function of random tilings on essentially arbitrary simply-connected domains of diameter N have magnitude smaller than N 1 / 2 . [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
27. Matrix Concentration for Products.
- Author
-
Huang, De, Niles-Weed, Jonathan, Tropp, Joel A., and Ward, Rachel
- Subjects
- *
MATRIX multiplications , *RANDOM matrices , *LARGE deviations (Mathematics) - Abstract
This paper develops nonasymptotic growth and concentration bounds for a product of independent random matrices. These results sharpen and generalize recent work of Henriksen–Ward, and they are similar in spirit to the results of Ahlswede–Winter and of Tropp for a sum of independent random matrices. The argument relies on the uniform smoothness properties of the Schatten trace classes. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
28. Orthogonal and Multiple Orthogonal Polynomials, Random Matrices, and Painlevé Equations
- Author
-
Van Assche, Walter, Foupouagnigni, Mama, editor, and Koepf, Wolfram, editor
- Published
- 2020
- Full Text
- View/download PDF
29. Stability-instability transition in tripartite merged ecological networks.
- Author
-
Emary, Clive and Malchow, Anne-Kathleen
- Subjects
- *
PHASE transitions , *BIPARTITE graphs , *RANDOM matrices , *NUMBERS of species , *FOOD chains - Abstract
Although ecological networks are typically constructed based on a single type of interaction, e.g. trophic interactions in a food web, a more complete picture of ecosystem composition and functioning arises from merging networks of multiple interaction types. In this work, we consider tripartite networks constructed by merging two bipartite networks, one mutualistic and one antagonistic. Taking the interactions within each sub-network to be distributed randomly, we consider the stability of the dynamics of the network based on the spectrum of its community matrix. In the asymptotic limit of a large number of species, we show that the spectrum undergoes an eigenvalue phase transition, which leads to an abrupt destabilisation of the network as the ratio of mutualists to antagonists is increased. We also derive results that show how this transition is manifest in networks of finite size, as well as when disorder is introduced in the segregation of the two interaction types. Our random-matrix results will serve as a baseline for understanding the behaviour of merged networks with more realistic structures and/or more detailed dynamics. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
30. Strong Solutions to a Beta-Wishart Particle System.
- Author
-
Jourdain, Benjamin and Kahn, Ezéchiel
- Abstract
The purpose of this paper is to study the existence and uniqueness of solutions to a stochastic differential equation (SDE) coming from the eigenvalues of Wishart processes. The coordinates are non-negative, evolve as Cox–Ingersoll–Ross (CIR) processes and repulse each other according to a Coulombian like interaction force. We show the existence of strong and pathwise unique solutions to the system until the first multiple collision and give a necessary and sufficient condition on the parameters of the SDEs for this multiple collision not to occur in finite time. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
31. On the smallest singular value of symmetric random matrices.
- Author
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Jain, Vishesh, Sah, Ashwin, and Sawhney, Mehtaab
- Subjects
SYMMETRIC matrices ,RANDOM matrices ,GEOMETRIC approach ,RANDOM variables ,ARITHMETIC ,RANDOM graphs ,TECHNOLOGICAL innovations - Abstract
We show that for an $n\times n$ random symmetric matrix $A_n$ , whose entries on and above the diagonal are independent copies of a sub-Gaussian random variable $\xi$ with mean 0 and variance 1, \begin{equation*}\mathbb{P}[s_n(A_n) \le \epsilon/\sqrt{n}] \le O_{\xi}(\epsilon^{1/8} + \exp(\!-\Omega_{\xi}(n^{1/2}))) \quad \text{for all } \epsilon \ge 0.\end{equation*} This improves a result of Vershynin, who obtained such a bound with $n^{1/2}$ replaced by $n^{c}$ for a small constant c, and $1/8$ replaced by $(1/8) - \eta$ (with implicit constants also depending on $\eta > 0$). Furthermore, when $\xi$ is a Rademacher random variable, we prove that \begin{equation*}\mathbb{P}[s_n(A_n) \le \epsilon/\sqrt{n}] \le O(\epsilon^{1/8} + \exp(\!-\Omega((\!\log{n})^{1/4}n^{1/2}))) \quad \text{for all } \epsilon \ge 0.\end{equation*} The special case $\epsilon = 0$ improves a recent result of Campos, Mattos, Morris, and Morrison, which showed that $\mathbb{P}[s_n(A_n) = 0] \le O(\exp(\!-\Omega(n^{1/2}))).$ Notably, in a departure from the previous two best bounds on the probability of singularity of symmetric matrices, which had relied on somewhat specialized and involved combinatorial techniques, our methods fall squarely within the broad geometric framework pioneered by Rudelson and Vershynin, and suggest the possibility of a principled geometric approach to the study of the singular spectrum of symmetric random matrices. The main innovations in our work are new notions of arithmetic structure – the Median Regularized Least Common Denominator (MRLCD) and the Median Threshold, which are natural refinements of the Regularized Least Common Denominator (RLCD)introduced by Vershynin, and should be more generally useful in contexts where one needs to combine anticoncentration information of different parts of a vector. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
32. Phase transition of eigenvalues in deformed Ginibre ensembles I: GinUE.
- Author
-
Liu, Dang-Zheng and Zhang, Lu
- Subjects
- *
RANDOM matrices , *PHASE transitions , *EIGENVALUES , *POINT processes , *MATHEMATICS - Abstract
Consider a random matrix of size
N as an additive deformation of the complex Ginibre ensemble under a deterministic matrix X0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$X_0$$\end{document} with a finite rank, independent ofN . When some eigenvalues of X0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$X_0$$\end{document} separate from the unit disk, outlier eigenvalues may appear asymptotically in the same locations, and their fluctuations exhibit surprising phenomena that highly depend on the Jordan canonical form of X0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$X_0$$\end{document}. These findings are largely due to Benaych-Georges and Rochet (Probab. Theory Relat. Fields, 165:313–363, 2016), Bordenave and Capitaine (Comm. Pure Appl. Math. 69:2131–2194, 2016), and Tao (Probab. Theory Relat. Fields 155:231–263, 2013). Yet when there is an eigenvalue of X0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$X_0$$\end{document} on the edge of the unit disk, we prove that local eigenvalue statistics at the same spectral edge form a new class of determinantal point processes, for which correlation kernels only depend on geometric multiplicity of eigenvalue and are characterized in terms of the iterated erfc functions. This thus completes a non-Hermitian analogue of the BBP phase transition in Random Matrix Theory. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
33. Phase transition for the smallest eigenvalue of covariance matrices.
- Author
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Bao, Zhigang, Lee, Jaehun, and Xu, Xiaocong
- Subjects
- *
COVARIANCE matrices , *PHASE transitions , *EIGENVALUES , *ASYMPTOTIC expansions , *RANDOM matrices , *POWER law (Mathematics) - Abstract
In this paper, we study the smallest non-zero eigenvalue of the sample covariance matrices S(Y)=YY∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {S}(Y)=YY^*$$\end{document}, where Y=(yij)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Y=(y_{ij})$$\end{document} is an M×N\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M\times N$$\end{document} matrix with iid mean 0 variance N-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N^{-1}$$\end{document} entries. We consider the regime M=M(N)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M=M(N)$$\end{document} and M/N→c∞∈R\{1}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M/N\rightarrow c_\infty \in \mathbb {R}{\setminus } \{1\}$$\end{document} as N→∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N\rightarrow \infty $$\end{document}. It is known that for the extreme eigenvalues of Wigner matrices and the largest eigenvalue of S(Y)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {S}(Y)$$\end{document}, a weak 4th moment condition is necessary and sufficient for the Tracy–Widom law (Ding and Yang in Ann Appl Probab 28(3):1679–1738, 2018. https://doi.org/10.1214/17-AAP1341; Lee and Yin in Duke Math J 163(1):117–173, 2014. https://doi.org/10.1215/00127094-2414767). In this paper, we show that the Tracy–Widom law is more robust for the smallest eigenvalue of S(Y)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {S}(Y)$$\end{document}, by discovering a phase transition induced by the fatness of the tail of yij\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$y_{ij}$$\end{document}’s. More specifically, we assume that yij\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$y_{ij}$$\end{document} is symmetrically distributed with tail probability P(|Nyij|≥x)∼x-α\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {P}(|\sqrt{N}y_{ij}|\ge x)\sim x^{-\alpha }$$\end{document} when x→∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x\rightarrow \infty $$\end{document}, for some α∈(2,4)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \in (2,4)$$\end{document}. We show the following conclusions: (1) When α>83\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha >\frac{8}{3}$$\end{document}, the smallest eigenvalue follows the Tracy–Widom law on scale N-23\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N^{-\frac{2}{3}}$$\end{document}; (2) When 2<α<83\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$2<\alpha <\frac{8}{3}$$\end{document}, the smallest eigenvalue follows the Gaussian law on scale N-α4\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N^{-\frac{\alpha }{4}}$$\end{document}; (3) When α=83\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha =\frac{8}{3}$$\end{document}, the distribution is given by an interpolation between Tracy–Widom and Gaussian; (4) In case α≤103\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\alpha \le \frac{10}{3}$$\end{document}, in addition to the left edge of the MP law, a deterministic shift of order N1-α2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N^{1-\frac{\alpha }{2}}$$\end{document} shall be subtracted from the smallest eigenvalue, in both the Tracy–Widom law and the Gaussian law. Overall speaking, our proof strategy is inspired by Aggarwal et al. (J Eur Math Soc 23(11):3707–3800, 2021. https://doi.org/10.4171/jems/1089) which is originally done for the bulk regime of the Lévy Wigner matrices. In addition to various technical complications arising from the bulk-to-edge extension, two ingredients are needed for our derivation: an intermediate left edge local law based on a simple but effective matrix minor argument, and a mesoscopic CLT for the linear spectral statistic with asymptotic expansion for its expectation. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
34. Fluctuations of β-Jacobi product processes.
- Author
-
Ahn, Andrew
- Subjects
- *
POINT processes , *DETERMINISTIC processes , *MARKOV processes , *RANDOM matrices - Abstract
We study Markov chains formed by squared singular values of products of truncated orthogonal, unitary, symplectic matrices (corresponding to the Dyson index β = 1 , 2 , 4 respectively) where time corresponds to the number of terms in the product. More generally, we consider the β -Jacobi product process obtained by extrapolating to arbitrary β > 0 . For fixed time (i.e. number of factors is constant), we show that the global fluctuations are jointly Gaussian with explicit covariances. For time growing linearly with matrix size, we show convergence of moments after suitable rescaling. When β = 2 , our results imply that the right edge converges to a process which interpolates between the Airy point process and a deterministic configuration. This process connects a time-parametrized family of point processes appearing in the works of Akemann–Burda–Kieburg and Liu–Wang–Wang across time. In the arbitrary β > 0 case, our results show tightness of the particles near the right edge. The limiting moment formulas correspond to expressions for the Laplace transform of a conjectural β -generalization of the interpolating process. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
35. A ribbon graph derivation of the algebra of functional renormalization for random multi-matrices with multi-trace interactions.
- Author
-
Pérez-Sánchez, Carlos I.
- Abstract
We focus on functional renormalization for ensembles of several (say n ≥ 1 ) random matrices, whose potentials include multi-traces, to wit, the probability measure contains factors of the form exp [ - Tr (V 1) × ⋯ × Tr (V k) ] for certain noncommutative polynomials V 1 , … , V k ∈ C ⟨ n ⟩ in the n matrices. This article shows how the “algebra of functional renormalization”—that is, the structure that makes the renormalization flow equation computable—is derived from ribbon graphs, only by requiring the one-loop structure that such equation (due to Wetterich) is expected to have. Whenever it is possible to compute the renormalization flow in terms of U (N) -invariants, the structure gained is the matrix algebra M n (A n , N , ⋆) with entries in A n , N = (C ⟨ n ⟩ ⊗ C ⟨ n ⟩) ⊕ (C ⟨ n ⟩ ⊠ C ⟨ n ⟩) , being C ⟨ n ⟩ the free algebra generated by the n Hermitian matrices of size N (the flowing random variables) with multiplication of homogeneous elements in A n , N given, for each P , Q , U , W ∈ C ⟨ n ⟩ , by (U ⊗ W) ⋆ (P ⊗ Q) = P U ⊗ W Q , (U ⊠ W) ⋆ (P ⊗ Q) = U ⊠ P W Q , (U ⊗ W) ⋆ (P ⊠ Q) = W P U ⊠ Q , (U ⊠ W) ⋆ (P ⊠ Q) = Tr (W P) U ⊠ Q ,
which, together with the condition (λ U) ⊠ W = U ⊠ (λ W) for each complex λ , fully define the symbol ⊠ . [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
36. Convergence of the spectral radius of a random matrix through its characteristic polynomial.
- Author
-
Bordenave, Charles, Chafaï, Djalil, and García-Zelada, David
- Subjects
- *
RANDOM matrices , *ANALYTIC functions , *POLYNOMIALS , *GAUSSIAN function , *CENTRAL limit theorem , *RADIUS (Geometry) , *SQUARE root - Abstract
Consider a square random matrix with independent and identically distributed entries of mean zero and unit variance. We show that as the dimension tends to infinity, the spectral radius is equivalent to the square root of the dimension in probability. This result can also be seen as the convergence of the support in the circular law theorem under optimal moment conditions. In the proof we establish the convergence in law of the reciprocal characteristic polynomial to a random analytic function outside the unit disc, related to a hyperbolic Gaussian analytic function. The proof is short and differs from the usual approaches for the spectral radius. It relies on a tightness argument and a joint central limit phenomenon for traces of fixed powers. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
37. Random integral matrices: universality of surjectivity and the cokernel.
- Author
-
Nguyen, Hoi H. and Wood, Melanie Matchett
- Subjects
- *
RANDOM matrices , *SPARSE matrices , *LINEAR operators , *SURJECTIONS , *ABELIAN groups , *FINITE groups - Abstract
For a random matrix of entries sampled independently from a fairly general distribution in Z we study the probability that the cokernel is isomorphic to a given finite abelian group, or when it is cyclic. This includes the probability that the linear map between the integer lattices given by the matrix is surjective. We show that these statistics are asymptotically universal (as the size of the matrix goes to infinity), given by precise formulas involving zeta values, and agree with distributions defined by Cohen and Lenstra, even when the distribution of matrix entries is very distorted. Our method is robust and works for Laplacians of random digraphs and sparse matrices with the probability of an entry non-zero only n - 1 + ε . [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
38. Expected Centre of Mass of the Random Kodaira Embedding.
- Author
-
Hashimoto, Yoshinori
- Abstract
Let X ⊂ P N - 1 be a smooth projective variety. To each g ∈ S L (N , C) which induces the embedding g · X ⊂ P N - 1 given by the ambient linear action we can associate a matrix μ ¯ X (g) called the centre of mass, which depends nonlinearly on g. With respect to the probability measure on S L (N , C) induced by the Haar measure and the Gaussian unitary ensemble, we prove that the expectation of the centre of mass is a constant multiple of the identity matrix for any smooth projective variety. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
39. Eigenvector distribution in the critical regime of BBP transition.
- Author
-
Bao, Zhigang and Wang, Dong
- Subjects
- *
RANDOM matrices , *PHASE transitions , *EIGENVECTORS , *EIGENVALUES , *POINT processes - Abstract
In this paper, we study the random matrix model of Gaussian Unitary Ensemble (GUE) with fixed-rank (aka spiked) external source. We will focus on the critical regime of the Baik–Ben Arous–Péché (BBP) phase transition and establish the distribution of the eigenvectors associated with the leading eigenvalues. The distribution is given in terms of a determinantal point process with extended Airy kernel. Our result can be regarded as an eigenvector counterpart of the BBP eigenvalue phase transition [6]. The derivation of the distribution makes use of the recently re-discovered eigenvector–eigenvalue identity, together with the determinantal point process representation of the GUE minor process with external source. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
40. Singularity of sparse random matrices: simple proofs.
- Author
-
Ferber, Asaf, Kwan, Matthew, and Sauermann, Lisa
- Subjects
SPARSE matrices ,RANDOM matrices ,PROBABILITY theory - Abstract
Consider a random $n\times n$ zero-one matrix with 'sparsity' p, sampled according to one of the following two models: either every entry is independently taken to be one with probability p (the 'Bernoulli' model) or each row is independently uniformly sampled from the set of all length-n zero-one vectors with exactly pn ones (the 'combinatorial' model). We give simple proofs of the (essentially best-possible) fact that in both models, if $\min(p,1-p)\geq (1+\varepsilon)\log n/n$ for any constant $\varepsilon>0$ , then our random matrix is nonsingular with probability $1-o(1)$. In the Bernoulli model, this fact was already well known, but in the combinatorial model this resolves a conjecture of Aigner-Horev and Person. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
41. Differentiability of Stochastic Differential Equation Driven by d-Dimensional G-Brownian Motion with Respect to the Initial Data.
- Author
-
Bougherra, Rania, Boutabia, Hacène, and Belksier, Manel
- Subjects
- *
STOCHASTIC differential equations , *WIENER processes , *RANDOM matrices - Abstract
The present paper is devoted to the study of the differentiability of solutions of stochastic differential equations driven by d-dimensional G-Brownian motion with respect to the initial data. Matricial stochastic differential equation of derivative and its inverse are given. This extends results obtained by Lin in 2013, in the one-dimensional case. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
42. Scattering in Quantum Dots via Noncommutative Rational Functions.
- Author
-
Erdős, László, Krüger, Torben, and Nemish, Yuriy
- Subjects
- *
QUANTUM dots , *QUANTUM scattering , *RANDOM matrices , *DEGREES of freedom , *SPECTRAL energy distribution , *SQUARE root - Abstract
In the customary random matrix model for transport in quantum dots with M internal degrees of freedom coupled to a chaotic environment via N ≪ M channels, the density ρ of transmission eigenvalues is computed from a specific invariant ensemble for which explicit formula for the joint probability density of all eigenvalues is available. We revisit this problem in the large N regime allowing for (i) arbitrary ratio ϕ : = N / M ≤ 1 ; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit ϕ → 0 , we recover the formula for the density ρ that Beenakker (Rev Mod Phys 69:731–808, 1997) has derived for a special matrix ensemble. We also prove that the inverse square root singularity of the density at zero and full transmission in Beenakker's formula persists for any ϕ < 1 but in the borderline case ϕ = 1 an anomalous λ - 2 / 3 singularity arises at zero. To access this level of generality, we develop the theory of global and local laws on the spectral density of a large class of noncommutative rational expressions in large random matrices with i.i.d. entries. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
43. Uniqueness of Dirichlet Forms Related to Infinite Systems of Interacting Brownian Motions.
- Author
-
Kawamoto, Yosuke, Osada, Hirofumi, and Tanemura, Hideki
- Abstract
The Dirichlet forms related to various infinite systems of interacting Brownian motions are studied. For a given random point field μ, there exist two natural infinite-volume Dirichlet forms (E u p r , D u p r ) and (E l w r , D l w r ) on L
2 (S,μ) describing interacting Brownian motions each with unlabeled equilibrium state μ. The former is a decreasing limit of a scheme of such finite-volume Dirichlet forms, and the latter is an increasing limit of another scheme of such finite-volume Dirichlet forms. Furthermore, the latter is an extension of the former. We present a sufficient condition such that these two Dirichlet forms are the same. In the first main theorem (Theorem 3.1) the Markovian semi-group given by (E l w r , D l w r ) is associated with a natural infinite-dimensional stochastic differential equation (ISDE). In the second main theorem (Theorem 3.2), we prove that these Dirichlet forms coincide with each other by using the uniqueness of weak solutions of ISDE. We apply Theorem 3.1 to stochastic dynamics arising from random matrix theory such as the sine, Bessel, and Ginibre interacting Brownian motions and interacting Brownian motions with Ruelle's class interaction potentials, and Theorem 3.2 to the sine2 interacting Brownian motion and interacting Brownian motions with Ruelle's class interaction potentials of C 0 3 -class. [ABSTRACT FROM AUTHOR]- Published
- 2021
- Full Text
- View/download PDF
44. Harry Kesten's work in probability theory.
- Author
-
Grimmett, Geoffrey R.
- Subjects
- *
PROBABILITY theory , *RANDOM walks , *PERCOLATION , *BRANCHING processes , *RANDOM matrices - Abstract
We survey the published work of Harry Kesten in probability theory, with emphasis on his contributions to random walks, branching processes, percolation, and related topics. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
45. Infinite-Dimensional Stochastic Differential Equations with Symmetry
- Author
-
Osada, Hirofumi, Eberle, Andreas, editor, Grothaus, Martin, editor, Hoh, Walter, editor, Kassmann, Moritz, editor, Stannat, Wilhelm, editor, and Trutnau, Gerald, editor
- Published
- 2018
- Full Text
- View/download PDF
46. Fluctuations of extreme eigenvalues of sparse Erdős–Rényi graphs.
- Author
-
He, Yukun and Knowles, Antti
- Subjects
- *
EIGENVALUES , *SPARSE matrices , *RANDOM matrices , *RANDOM graphs , *RANDOM variables , *MATHEMATICS - Abstract
We consider a class of sparse random matrices which includes the adjacency matrix of the Erdős–Rényi graph G (N , p) . We show that if N ε ⩽ N p ⩽ N 1 / 3 - ε then all nontrivial eigenvalues away from 0 have asymptotically Gaussian fluctuations. These fluctuations are governed by a single random variable, which has the interpretation of the total degree of the graph. This extends the result (Huang et al. in Ann Prob 48:916–962, 2020) on the fluctuations of the extreme eigenvalues from N p ⩾ N 2 / 9 + ε down to the optimal scale N p ⩾ N ε . The main technical achievement of our proof is a rigidity bound of accuracy N - 1 / 2 - ε (N p) - 1 / 2 for the extreme eigenvalues, which avoids the (N p) - 1 -expansions from Erdős et al. (Ann Prob 41:2279–2375, 2013), Huang et al. (2020) and Lee and Schnelli (Prob Theor Rel Fields 171:543–616, 2018). Our result is the last missing piece, added to Erdős et al. (Commun Math Phys 314:587–640, 2012), He (Bulk eigenvalue fluctuations of sparse random matrices. arXiv:1904.07140), Huang et al. (2020) and Lee and Schnelli (2018), of a complete description of the eigenvalue fluctuations of sparse random matrices for N p ⩾ N ε . [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
47. Diffusions interacting through a random matrix: universality via stochastic Taylor expansion.
- Author
-
Dembo, Amir and Gheissari, Reza
- Subjects
- *
RANDOM matrices , *TAYLOR'S series , *RANDOM dynamical systems , *STOCHASTIC differential equations , *HOPFIELD networks - Abstract
Consider (X i (t)) solving a system of N stochastic differential equations interacting through a random matrix J = (J ij) with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of (X i (t)) , initialized from some μ independent of J , are universal, i.e., only depend on the choice of the distribution J through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
48. Function Values Are Enough for L2-Approximation.
- Author
-
Krieg, David and Ullrich, Mario
- Subjects
- *
SOBOLEV spaces , *HILBERT functions , *ALGORITHMS , *POLYNOMIAL approximation , *HILBERT space , *SMOOTHNESS of functions - Abstract
We study the L 2 -approximation of functions from a Hilbert space and compare the sampling numbers with the approximation numbers. The sampling number e n is the minimal worst-case error that can be achieved with n function values, whereas the approximation number a n is the minimal worst-case error that can be achieved with n pieces of arbitrary linear information (like derivatives or Fourier coefficients). We show that e n ≲ 1 k n ∑ j ≥ k n a j 2 , where k n ≍ n / log (n) . This proves that the sampling numbers decay with the same polynomial rate as the approximation numbers and therefore that function values are basically as powerful as arbitrary linear information if the approximation numbers are square-summable. Our result applies, in particular, to Sobolev spaces H mix s (T d) with dominating mixed smoothness s > 1 / 2 and dimension d ∈ N , and we obtain e n ≲ n - s log sd (n). For d > 2 s + 1 , this improves upon all previous bounds and disproves the prevalent conjecture that Smolyak's (sparse grid) algorithm is optimal. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
49. Sharp transition of the invertibility of the adjacency matrices of sparse random graphs.
- Author
-
Basak, Anirban and Rudelson, Mark
- Subjects
- *
SPARSE matrices , *BIPARTITE graphs , *UNDIRECTED graphs , *SPARSE graphs , *RANDOM matrices , *RANDOM graphs , *MATRICES (Mathematics) - Abstract
We consider three models of sparse random graphs: undirected and directed Erdős–Rényi graphs and random bipartite graph with two equal parts. For such graphs, we show that if the edge connectivity probability p satisfies n p ≥ log n + k (n) with k (n) → ∞ as n → ∞ , then the adjacency matrix is invertible with probability approaching one (n is the number of vertices in the two former cases and the same for each part in the latter case). For n p ≤ log n - k (n) these matrices are invertible with probability approaching zero, as n → ∞ . In the intermediate region, when n p = log n + k (n) , for a bounded sequence k (n) ∈ R , the event Ω 0 that the adjacency matrix has a zero row or a column and its complement both have a non-vanishing probability. For such choices of p our results show that conditioned on the event Ω 0 c the matrices are again invertible with probability tending to one. This shows that the primary reason for the non-invertibility of such matrices is the existence of a zero row or a column. We further derive a bound on the (modified) condition number of these matrices on Ω 0 c , with a large probability, establishing von Neumann's prediction about the condition number up to a factor of n o (1) . [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
50. VICTORIA transform, RESPECT and REFORM methods for the proof of the G-permanent pencil law under G-Lindeberg condition for some random matrices from G-elliptic ensemble.
- Author
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Girko, Vyacheslav L.
- Subjects
- *
RANDOM matrices , *EVIDENCE , *PENCILS , *REFORMS , *RESPECT - Abstract
The G-pencil law under the G-Lindeberg condition for a random matrix is proven. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
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