Journal ArticleDOI
Non-Hermitian Tridiagonal Random Matrices and Returns to the Origin of a Random Walk
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In this paper, a tridiagonal matrix model, the q-root of unity model, was studied and the eigenvalue densities were bounded by and have the symmetries of the regular polygon with 2q sides in the complex plane.Abstract:
We study a class of tridiagonal matrix models, the “q-roots of unity” models, which includes the sign (q=2) and the clock (q=∞) models by Feinberg and Zee. We find that the eigenvalue densities are bounded by and have the symmetries of the regular polygon with 2q sides, in the complex plane. Furthermore, the averaged traces of Mk are integers that count closed random walks on the line such that each site is visited a number of times multiple of q. We obtain an explicit evaluation for them.read more
Citations
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A Complete Bibliography of the Journal of Statistical Physics: 2000{2009
TL;DR: In this paper, Zuc11b et al. this paper showed that 1 ≤ p ≤ ∞ [Dud13]. 1/f [HPF15], 1/n [Per17] and 1/m [DFL17] were the most frequent p ≤ p ≥ ∞.
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On the Spectra and Pseudospectra of a Class of Non-Self-Adjoint Random Matrices and Operators
TL;DR: In this article, the authors developed and applied methods for the spectral analysis of non-self-adjoint tridiagonal innite and nite random matrices, and also proposed a sequence of inclusion sets for which they show is convergent to, with the nth element of the sequence computable by calculating smallest singular values of (large numbers of) n n matrices.
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Spectrum of a Feinberg-Zee random hopping matrix
TL;DR: In this paper, a new proof of a theorem of Chandler-Wilde, Chonchaiya, and Lindner that the spectra of a certain class of infinite, random, tridiagonal matrices contain the unit disc almost surely was provided.
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On the remarkable spectrum of a non-Hermitian random matrix model
TL;DR: Using the Dyson?Schmidt equation, this paper showed that the spectrum of a non-denumerable set of lines in the complex plane is the support of a periodic Hamiltonian, obtained by the infinite repetition of any finite sequence of the disorder variables.
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Enumeration of simple random walks and tridiagonal matrices
TL;DR: The relation between the trace of the nth power of a tridiagonal matrix and the enumeration of weighted paths of n steps allows an easier combinatorial enumeration.
References
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Journal ArticleDOI
Random-matrix theory of quantum transport
TL;DR: In this article, a review of the statistical properties of the scattering matrix of a mesoscopic system is presented, where two geometries are contrasted: a quantum dot and a disordered wire.
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Localization Transitions in Non-Hermitian Quantum Mechanics.
Naomichi Hatano,David R. Nelson +1 more
TL;DR: The theory predicts that, close to the depinning transition, the transverse Meissner effect is accompanied by stretched exponential relaxation of the field into the bulk and a diverging penetration depth.
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Spectrum of large random asymmetric matrices
TL;DR: It is found that $\ensuremath{\rho}(\ensure Math{\lambda})$ is uniform in an ellipse, in the complex plane, whose real and imaginary axes are $1+\ensureMath{\tau}$ and $1\ensuresuremath{-}\ensure maths{\infty}$, respectively.
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Distribution of Eigenvalues in Non-Hermitian Anderson Models
TL;DR: In this article, the authors developed a theory which describes the behavior of eigenvalues of a class of one-dimensional random non-Hermitian operators introduced by Hatano and Nelson.
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Density of states in the non-hermitian lloyd model
TL;DR: In this paper, it was shown that it is possible to find the localization length from the density of states of a non-Hermitian random Hamiltonian, and that the problem of finding the density in a nonhermitian Hamiltonian remains open.