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580
On the spheredecoding algorithm I. Expected complexity
 IEEE Trans. Sig. Proc
, 2005
"... Abstract—The problem of finding the leastsquares solution to a system of linear equations where the unknown vector is comprised of integers, but the matrix coefficient and given vector are comprised of real numbers, arises in many applications: communications, cryptography, GPS, to name a few. The ..."
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Cited by 76 (5 self)
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Abstract—The problem of finding the leastsquares solution to a system of linear equations where the unknown vector is comprised of integers, but the matrix coefficient and given vector are comprised of real numbers, arises in many applications: communications, cryptography, GPS, to name a few. The problem is equivalent to finding the closest lattice point to a given point and is known to be NPhard. In communications applications, however, the given vector is not arbitrary but rather is an unknown lattice point that has been perturbed by an additive noise vector whose statistical properties are known. Therefore, in this paper, rather than dwell on the worstcase complexity of the integer leastsquares problem, we study its expected complexity, averaged over the noise and over the lattice. For the “sphere decoding” algorithm of Fincke and Pohst, we find a closedform expression for the expected complexity, both for the infinite and finite lattice.
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 (8 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
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,
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 56 (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.
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 56 (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.
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 55 (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...
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 48 (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].