On Slow-Fading MIMO Systems With Nonseparable Correlation

Far, Reza Rashidi; Oraby, Tamer; Bryc, Włodzimierz; Speicher, Roland

doi:10.1109/tit.2007.913239

Cited by 54 publications

(46 citation statements)

References 25 publications

(41 reference statements)

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“…Special correlation structures were also considered before in [1, 11] on a local scale. We also mention that there exists an extensive literature on the global law for random matrices with correlated entries [6,8,9,25,26,39,40]. These results, however, either concern Gaussian random matrices or more specific correlation structures than considered in the present work.…”

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confidence: 85%

Random Matrices With Slow Correlation Decay

Erdős

Krüger

Schröder

2019

Forum of Mathematics, Sigma

203

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We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the recent result of [3] to allow slow correlation decay and arbitrary expectation. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion. ContentsRANDOM MATRICES WITH SLOW CORRELATION DECAY 2 of the individual eigenvalues follows a universal distribution, independent of the specifics of the random matrix itself. The former is commonly called a local law, whereas the latter is known as the Wigner-Dyson-Mehta (WDM) universality conjecture, first envisioned by Wigner in the 1950's and formalized later by Dyson and Mehta in the 1960's [36]. In fact, the conjecture extends beyond the customary random matrix ensembles in probability theory and is believed to hold for any random operator in the delocalization regime of the Anderson metal-insulator phase transition. Given this profound universality conjecture for general disordered quantum systems, the ultimate goal of local spectral analysis of large random matrices is to prove the WDM conjecture for the largest possible class of matrix ensembles. In the current paper we complete this program for random matrices with a general, slow correlation decay among its matrix elements. Previous works covered only correlations with such a fast decay that, in a certain sense, they could be treated as a perturbation of the independent model. Here we present a new method that goes well beyond the perturbative regime. It relies on a novel multi-scale version of the cumulant expansion and its rigorous Feynman diagrammatic representation that can be useful for other problems as well. To put our work in context, we now explain the previous results.In the last ten years a powerful new approach, the three-step strategy has been developed to resolve WDM universality problems, see [19] for a summary. In particular, the WDM conjecture in its classical form, stated for Wigner matrices with a general distribution of the entries, has been proven with this strategy in [14,15,21]; an independent proof for the Hermitian symmetry class was given in [42]. Recent advances have crystallized that the only model dependent step in this strategy is the first one, the local law. The other two steps, the fast relaxation to equilibrium of the Dyson Brownian motion and the approximation by Gaussian divisible ensembles, have been formulated as very general "black-box" tools whose only input is the local law [17,31,32]. Thus the proof of the WDM universality, at least for mean field ensembles, is automatically reduced to obtaining a local law.Both local law and universality have first been established for Wigner matrices, which are real symmetric or complex Hermitian N × N matrices with mean-zero entries which are independent and identically distributed (i.i.d.) up to symmetry [15,16]. For Wigner matrices it has long been known that the l...

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confidence: 85%

Random Matrices With Slow Correlation Decay

Erdős

Krüger

Schröder

2019

Forum of Mathematics, Sigma

203

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“…There are few papers however covering the behavior of the ESD of band matrices with dependent entries. Some of the best known results can be found in [Sly96] and [ROBS08]. The last paper deals with a wider class of block matrices.…”

Section: 4mentioning

confidence: 99%

Concentration of the empirical spectral distribution of random matrices with dependent entries

Polaczyk¹

2019

Electron. Commun. Probab.

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We investigate concentration properties of spectral measures of Hermitian random matrices with partially dependent entries. More precisely, let Xn be a Hermitian random matrix of size n × n that can be split into independent blocks of the size at most dn = o(n 2 ). We prove that under some mild conditions on the distribution of the entries of Xn, the empirical spectral measure of Xn concentrates around its mean.The main theorem is a strengthening of a recent result by Kemp and Zimmerman, where the size of blocks grows as o(log n). As an application, we are able to upgrade the results of Schenker and Schulz on the convergence in expectation to the semicircle law of a class of random matrices with dependent entries to weak convergence in probability. Other applications include patterned random matrices, e.g. matrices of Toeplitz, Hankel or circulant type and matrices with heavy tailed entries in the domain of attraction of the Gaussian distribution.2010 Mathematics Subject Classification. 60B20.

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“…Even though this channel is of interest in itself, its main role is to serve as the basic building block for the main application of this paper, the compound channels. The random matrix model of the block Gaussian channels, which includes the random matrix model in [4], has been already used in Rashidi Far et al [18] in the context of frequency-selective slow-fading multiantenna channels. In that context, no numerical optimization was required.…”

Section: A Block Gaussian Channelsmentioning

confidence: 99%

“…In this direction, Shlyakhtenko [17] showed how to compute the asymptotic eigenvalue distribution of band Gaussian matrices, i.e., the random matrices associated to Gaussian IND channels, using operator-valued free probability theory; from which the computation of the asymptotic capacity follows by a standard argument [4]. These free probability approaches have also been applied in other contexts, such as frequency-selective slowfading channels, where Rashidi Far et al [18] approximated the capacity of a channel with an associated block random matrix in which an arbitrary correlation between blocks was allowed. Their research is heavily based on the free probability tools in Helton et al [19].…”

Section: Introductionmentioning

confidence: 99%

“…A situation where these techniques are particularly useful is in the study of sums of products of independent block Gaussian matrices (see Section III-B). If we start with independent matrices like the ones considered in [18] where an arbitrary correlation between blocks is allowed, then the entries of a sum of products of these Gaussian matrices are neither Gaussian, nor independent, nor identically distributed. A main feature of the operator-valued free probability approach resides in the systematic use of the concept of freeness introduced by Voiculescu [25], which allows us to compute the asymptotic eigenvalue distribution of algebraic operations (sums and products) of certain independent block random matrices.…”

Section: Introductionmentioning

confidence: 99%

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On the Capacity of Block Multiantenna Channels

Díaz

Pérez‐Abreu

2017

IEEE Trans. Inform. Theory

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Abstract-In this paper we consider point-to-point multiantenna channels with certain block distributional symmetries which do not require the entries of the channel matrix to be either Gaussian, or independent, or identically distributed. A main contribution is a capacity theorem for these channels, which might be regarded as a generalization of Telatar's theorem (1999), which reduces the numerical optimization domain in the capacity computation. With this information theoretic result and some free probability arguments, we prove an asymptotic capacity theorem that, in addition to reducing the optimization domain, does not depend on the dimension of the channel matrix. This theorem allows us to apply free probability techniques to numerically compute the asymptotic capacity of the channels under consideration. These theorems provide a very efficient method for numerically approximating both the capacity and a capacity achieving input covariance matrix of certain channels.

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On Slow-Fading MIMO Systems With Nonseparable Correlation

Cited by 54 publications

References 25 publications

Random Matrices With Slow Correlation Decay

Random Matrices With Slow Correlation Decay

Concentration of the empirical spectral distribution of random matrices with dependent entries

On the Capacity of Block Multiantenna Channels

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