Results 11  20
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580
A unified framework for highdimensional analysis of Mestimators with decomposable regularizers
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On the concentration of eigenvalues of random symmetric matrices
 Israel J. Math
, 2000
"... It is shown that for every 1 ≤ s ≤ n, the probability that the sth largest eigenvalue of a random symmetric nbyn matrix with independent random entries of absolute value at most 1 deviates from its median by more than t is at most 4e −t2 /32s 2. The main ingredient in the proof is Talagrand’s Ine ..."
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Cited by 62 (7 self)
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It is shown that for every 1 ≤ s ≤ n, the probability that the sth largest eigenvalue of a random symmetric nbyn matrix with independent random entries of absolute value at most 1 deviates from its median by more than t is at most 4e −t2 /32s 2. The main ingredient in the proof is Talagrand’s Inequality for concentration of measure in product spaces. 1
Limiting Distributions for a Polynuclear Growth Model With External Sources
, 2000
"... The purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources, which was considered by Pr ahofer and Spohn in [13]. Depending on the strength of the sources, the limiting distribution functions are either the TracyWidom func ..."
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Cited by 59 (9 self)
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The purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources, which was considered by Pr ahofer and Spohn in [13]. Depending on the strength of the sources, the limiting distribution functions are either the TracyWidom functions of random matrix theory, or a new explicit function which has the special property that its mean is zero. Moreover, we obtain transition functions between pairs of the above distribution functions in suitably scaled limits. There are also similar results for a discrete totally asymmetric exclusion process.
A Random Matrix Model of Communication via Antenna Arrays
, 2001
"... A random matrix model is introduced that probabilistically describes the spatial and temporal multipath propagation between a transmitting and receiving antenna array with a limited number of scatterers for mobile radio and indoor environments. The model characterizes the channel by its richness d ..."
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Cited by 57 (7 self)
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A random matrix model is introduced that probabilistically describes the spatial and temporal multipath propagation between a transmitting and receiving antenna array with a limited number of scatterers for mobile radio and indoor environments. The model characterizes the channel by its richness delay profile which gives the number of scattering objects as a function of the path delay. Each delay is assigned the eigenvalue distribution of a random matrix that depends on the number of scatterers, receive antennas, and transmit antennas. The model allows to calculate signaltointerferenceandnoise ratios and channel capacities for large antenna arrays analytically and quantifies up to what extent rich scattering improves performance.
Infinite wedge and random partitions
 Selecta Mathematica (new series
"... The aim of this paper is to show that random partitions have a very natural and direct connection to various structures which are well known in integrable systems. This connection is arguably even more natural than, for example, ..."
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Cited by 56 (6 self)
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The aim of this paper is to show that random partitions have a very natural and direct connection to various structures which are well known in integrable systems. This connection is arguably even more natural than, for example,
Universality Of The Local Eigenvalue Statistics For A Class Of Unitary Invariant Random Matrix Ensembles
, 1997
"... The paper is devoted to the rigorous proof of the universality conjecture of the random matrix theory, according to which the limiting eigenvalue statistics of n \Theta n random matrices within spectral intervals of the order O(n \Gamma1 ) is determined by the type of matrices (real symmetric, Her ..."
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Cited by 54 (4 self)
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The paper is devoted to the rigorous proof of the universality conjecture of the random matrix theory, according to which the limiting eigenvalue statistics of n \Theta n random matrices within spectral intervals of the order O(n \Gamma1 ) is determined by the type of matrices (real symmetric, Hermitian or quaternion real) and by the density of states. We prove this conjecture for a certain class of the Hermitian matrix ensembles that arose in the quantum field theory and have the unitary invariant distribution defined by a certain function (the potential in the quantum field theory) satisfying some regularity conditions. Key words: random matrices, local asymptotic regime, universality conjecture, orthogonal polynomial technique. 1 Introduction. Problem and results. The random matrix theory (RMT) has been extensively developed and used in a number of areas of theoretical and mathematical physics. In particular the theory provides quite satisfactory description of fluctuations in s...
Matrices coupled in a chain: eigenvalue correlations. Saclay preprint SPhT 97/112
"... Abstract. The general correlation function for the eigenvalues of p complex hermitian n × n matrices coupled in a chain is given as a single determinant. For this we use a slight generalization of a theorem of Dyson. 1. ..."
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Cited by 50 (15 self)
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Abstract. The general correlation function for the eigenvalues of p complex hermitian n × n matrices coupled in a chain is given as a single determinant. For this we use a slight generalization of a theorem of Dyson. 1.
The Asymptotics of Monotone Subsequences of Involutions
, 2001
"... We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involutions tend to infinity. The resulting distributions axe, depending on the ..."
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Cited by 50 (5 self)
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We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involutions tend to infinity. The resulting distributions axe, depending on the number of fixed points, (1) the TracyWidom distributions for the laxgest eigenvalues of random GOE, GUE, GSE matrices, (2) the normal distribution, or (3) new classes of distributions which interpolate between pairs of the Tracy Widom distributions. We also consider the second rows of the corresponding Young diagrams. In each case the convergence of moments is also shown. The proof is based on the algebraic work of the authors in [7] which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials, and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the RiemannHilbert analysis for the orthogonal polynomials by Delft, Johansson and the first author in [3].
How many zeros of a random polynomial are real
 Bull. Amer. Math. Soc. (N.S
, 1995
"... Abstract. We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve (1, t, ..."
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Cited by 46 (0 self)
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Abstract. We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve (1, t,..., t n) projected onto the surface of the unit sphere, divided by π. The probability density of the real zeros is proportional to how fast this curve is traced out. We then relax Kac’s assumptions by considering a variety of random sums, series, and distributions, and we also illustrate such ideas as integral geometry and the FubiniStudy metric. Contents 1.