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16
Universality in Polytope Phase Transitions and Message Passing Algorithms
, 2012
"... We consider a class of nonlinear mappings FA,N in R N indexed by symmetric random matrices A ∈ R N×N with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating the TAP equations and were studied by Erwin Bolthausen. Within information theory, they are ..."
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Cited by 24 (4 self)
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We consider a class of nonlinear mappings FA,N in R N indexed by symmetric random matrices A ∈ R N×N with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating the TAP equations and were studied by Erwin Bolthausen. Within information theory, they are known as ‘approximate message passing ’ algorithms. We study the highdimensional (large N) behavior of the iterates of F for polynomial functions F, and prove that it is universal, i.e. it depends only on the first two moments of the entries of A, under a subgaussian tail condition. As an application, we prove the universality of a certain phase transition arising in polytope geometry and compressed sensing. This solves –for a broad class of random projections – a conjecture by David Donoho and Jared Tanner. 1 Introduction and main results Let A ∈ RN×N be a random Wigner matrix, i.e. a random matrix with i.i.d. entries Aij satisfying E{Aij} = 0 and E{A2 ij} = 1/N. Considerable effort has been devoted to studying the distribution of the eigenvalues of such a matrix [AGZ09, BS05, TV12]. The universality phenomenon is a striking recurring theme in these studies. Roughly speaking, many asymptotic properties of the joint eigenvalues
Random matrices: Universality of local spectral statistics of nonHermitian matrices
, 2013
"... It is a classical result of Ginibre that the normalized bulk kpoint correlation functions of a complex n × n gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process on C with kernel K∞(z, w): = 1pi e −z2/2−w2/2+zw in ..."
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Cited by 11 (1 self)
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It is a classical result of Ginibre that the normalized bulk kpoint correlation functions of a complex n × n gaussian matrix with independent entries of mean zero and unit variance are asymptotically given by the determinantal point process on C with kernel K∞(z, w): = 1pi e −z2/2−w2/2+zw in the limit n→∞. In this paper we show that this asymptotic law is universal among all random n × n matrices Mn whose entries are jointly independent, exponentially decaying, have independent real and imaginary parts, and whose moments match that of the complex gaussian ensemble to fourth order. Analogous results at the edge of the spectrum are also obtained. As an application, we extend a central limit theorem for the number of eigenvalues of complex gaussian matrices in a small disk to these more general ensembles. These results are nonHermitian analogues of some recent universality results for Hermitian Wigner matrices. However, a key new difficulty arises in the nonHermitian case, due to the instability of the spectrum for such ma
Random matrices: Sharp concentration of eigenvalues
 DEPARTMENT OF MATHEMATICS, ZHEJIANG UNIVERSITY
, 2013
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Universality of local eigenvalue statistics in random matrices with external source arXiv:1308.1057
, 2013
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The Spectrum of Random Kernel Matrices: Universality Results for Rough and Varying Kernels
 Random Matrices: Theory and Applications
, 2013
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Free Probability Theory and Infinitely divisible distributions
, 2013
"... Elements in a noncommutative operator algebra can be regarded as noncommutative random variables from a probabilistic viewpoint. Such understanding has its origin in quantum theory. Theory of operator algebras focusing on the probabihstic aspect is called noncommutative probability theory. Noncommut ..."
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Elements in a noncommutative operator algebra can be regarded as noncommutative random variables from a probabilistic viewpoint. Such understanding has its origin in quantum theory. Theory of operator algebras focusing on the probabihstic aspect is called noncommutative probability theory. Noncommutative probability theory is divided into several directions. Some groups perform mathematical research, and others do physical research. The main focus of this article is free probability, a mathematical aspect of noncommutative probability. The name of free probability theory might sound strange for nonexperts. This name was chosen because free probability fits in the analysis of the free product of groups or algebras. $\mathbb{R}ee $ probability has been developed in terms of operator algebras to solve problems related to von Neumann algebras generated by free groups [HPOO, VDN92]. From a probabilistic aspect, when one considers random walks on free groups, free probability is useful to analyze the recurrence/transience of the random walks [W86].1 In addition, Voiculescu [V91] found that free probabihty has application to the analysis of the eigenvalues of random matrices (see also [HPOO, VDN92]). Why eigenvalues of random matrices interest researchers? The original motivation is to model the energy levels of nucleons of nuclei. Then subsequent studies revealed many relations of random matrices to other mathematics as well as physics, e.g. integrable systems (such as Peinlev\’e equations), the Riemann zeta function and representation theory [M04]. All these applications are based on the analysis of eigenvalue distributions of random matrices. In this article, we are going to present the basics of free probability, and then describe the summary of results obtained so far on freely infinitely divisible distributions, the author’s recent main subject. $A $ purpose of free probability is to analyze free convolution which describes the eigenvalue distribution of the sum of independent large random matrices. The set of freely infinitely divisible probability measures is the central subject associated to free convolution. *email:
An Update on Local Universality Limits for Correlation Functions Generated by Unitary Ensembles
, 2016
"... Abstract. We survey the current status of universality limits for mpoint correlation functions in the bulk and at the edge for unitary ensembles, primarily when the limiting kernels are Airy, Bessel, or Sine kernels. In particular, we consider underlying measures on compact intervals, and fixed an ..."
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Abstract. We survey the current status of universality limits for mpoint correlation functions in the bulk and at the edge for unitary ensembles, primarily when the limiting kernels are Airy, Bessel, or Sine kernels. In particular, we consider underlying measures on compact intervals, and fixed and varying exponential weights, as well as universality limits for a variety of orthogonal systems. The scope of the survey is quite narrow: we do not consider β ensembles for β = 2, nor general Hermitian matrices with independent entries, let alone more general settings. We include some open problems.
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Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
SMITH NORMAL FORM IN COMBINATORICS
, 2015
"... This paper surveys some combinatorial aspects of Smith normal form, and more generally, diagonal form. The discussion includes general algebraic properties and interpretations of Smith normal form, critical groups of graphs, and Smith normal form of random integer matrices. We then give some examp ..."
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This paper surveys some combinatorial aspects of Smith normal form, and more generally, diagonal form. The discussion includes general algebraic properties and interpretations of Smith normal form, critical groups of graphs, and Smith normal form of random integer matrices. We then give some examples of Smith normal form and diagonal form arising from (1) symmetric functions, (2) a result of Carlitz, Roselle, and Scoville, and (3) the Varchenko matrix of a hyperplane arrangement.