The Thouless formula for random non-Hermitian Jacobi matrices

Goldsheid, Ilya; Khoruzhenko, Boris A.

doi:10.1007/bf02775442

Cited by 23 publications

(25 citation statements)

References 19 publications

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“…Lemma 6.4 follows from [9, Lemma 11.2] and is based on Girko's original observation [21,22]. The lemma has appeared in a number of different forms; for example, see [11,Lemma 4.3] and [25]. Lemma 6.4 (Lemma 11.2 from [9]).…”

Section: Completing the Argumentmentioning

confidence: 99%

Products of Independent Elliptic Random Matrices

O’Rourke

Renfrew²,

Soshnikov

et al. 2015

J Stat Phys

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Abstract. For fixed m > 1, we study the product of m independent N × N elliptic random matrices as N tends to infinity. Our main result shows that the empirical spectral distribution of the product converges, with probability 1, to the m-th power of the circular law, regardless of the joint distribution of the mirror entries in each matrix. This leads to a new kind of universality phenomenon: the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves.Our result also generalizes earlier results of and O'Rourke-Soshnikov [44] concerning the product of independent iid random matrices.

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Section: Completing the Argumentmentioning

confidence: 99%

Products of Independent Elliptic Random Matrices

O’Rourke

Renfrew²,

Soshnikov

et al. 2015

J Stat Phys

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“…We use the theory of product of random matrices theory. For a general introduction to the aspects of the theory we use here, the reader may consult [25], [26], [35]- [37].…”

Section: Product Of Random Matricesmentioning

confidence: 99%

On Certain Large Random Hermitian Jacobi Matrices With Applications to Wireless Communications

Levy

Somekh

Shamai

et al. 2009

IEEE Trans. Inform. Theory

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In this paper we study the spectrum of certain large random Hermitian Jacobi matrices. These matrices are known to describe certain communication setups. In particular we are interested in an uplink cellular channel which models mobile users experiencing a soft-handoff situation under joint multicell decoding. Considering rather general fading statistics we provide a closed form expression for the per-cell sum-rate of this channel in high-SNR, when an intra-cell TDMA protocol is employed.Since the matrices of interest are tridiagonal, their eigenvectors can be considered as sequences with second order linear recurrence. Therefore, the problem is r educed to the study of the exponential growth of products of two by two matrices. For the case where K users are simultaneously active in each cell, we obtain a series of lower and upper bound on the high-SNR power offset of the per-cell sum-rate, which are considerably tighter than previously known bounds.

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“…When a i /b i > 0 and {(a i−1 , d i , b i )} is a sequence of i.i.d. random vectors with all moments finite, this tridiagonal random matrix is motivated by the non-Hermitian quantum mechanics of Hatano and Nelson (see [21] and references therein). In this case, it follows from Corollary 3.5 that the traces are approximated normally distributed.…”

Section: Applicationsmentioning

confidence: 99%

“…The non-symmetric model with a i /b i > 0 also arises in the non-Hermitian quantum mechanics of Hatano and Nelson, see e.g. [21] and references therein. Tridiagonal random matrix is also a basic model of random walks with random environment in chain graphs, interesting examples include the random birth-death Markov kernel proposed in [5,8], where a i−1 + d i + b i = 1 and {(a i−1 , d i , b i )} is an ergodic random field, and the random birth-death Q matrix recently studied in [23], where d i = −(a i−1 + b i ) and {a i } and {b i } are two sequences of strictly stationary ergodic processes.…”

Section: Introductionmentioning

confidence: 99%

Tridiagonal Random Matrix: Gaussian Fluctuations and Deviations

Deng

2016

J Theor Probab

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This paper is devoted to the Gaussian fluctuations and deviations of the traces of tridiagonal random matrices. Under quite general assumptions, we prove that the traces are approximately normally distributed. A Multi-dimensional central limit theorem is also obtained here. These results have several applications to various physical models and random matrix models, such as the Anderson model, the random birth-death Markov kernel, the random birth-death Q matrix and the β-Hermite ensemble. Furthermore, under an independent-and-identically-distributed condition, we also prove the large deviation principle as well as the moderate deviation principle for the traces.

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The Thouless formula for random non-Hermitian Jacobi matrices

Cited by 23 publications

References 19 publications

Products of Independent Elliptic Random Matrices

Products of Independent Elliptic Random Matrices

On Certain Large Random Hermitian Jacobi Matrices With Applications to Wireless Communications

Tridiagonal Random Matrix: Gaussian Fluctuations and Deviations

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