Spectral Properties of Non-Unitary Band Matrices

Hamza, Eman; Joye, Alain

doi:10.1007/s00023-014-0385-6

Cited by 2 publications

(1 citation statement)

References 39 publications

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“…By Balanced Random Quantum Walk, BRQW for short, we mean a quantum walk defined either on the cubic lattice Z d or on the homogeneous tree T 2d , where d ∈ N is half the coordination number in the latter case, such that the quantum mechanical transition amplitude between neighbouring sites has uniform modulus and random argument uniformly distributed on the torus. We briefly recall the basics about RQWs below; for more details, see [HJ1,HJ2] 2.1 Random quantum walks on T 2d Let T 2d denote the homogeneous tree of degree 2d of the free group F {a 1 ,...,a d } generated by the alphabet…”

Section: Balanced Random Quantum Walkmentioning

confidence: 99%

Lower bounds on the localisation length of balanced random quantum walks

Asch

Joye

2019

Lett Math Phys

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We consider the dynamical properties of Quantum Walks defined on the d-dimensional cubic lattice, or the homogeneous tree of coordination number 2d, with site dependent random phases, further characterised by transition probabilities between neighbouring sites equal to 1/(2d). We show that the localisation length for these Balanced Random Quantum Walks can be expressed as a combinatorial expression involving sums over weighted paths on the considered graph. This expression provides lower bounds on the localisation length by restriction to paths with weight 1, which allows us to prove the localisation length diverges on the tree as d 2 . On the cubic lattice, the method yields the lower bound 1/ ln(2) for all d, and allows us to bound the localisation length from below by the correlation length of self-avoiding walks computed at 1/(2d).

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Section: Balanced Random Quantum Walkmentioning

confidence: 99%

Lower bounds on the localisation length of balanced random quantum walks

Asch

Joye

2019

Lett Math Phys

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Density of states for random contractions

Joye¹

2017

J. Spectr. Theory

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We define a linear functional, the DOS functional, on spaces of holomorphic functions on the unit disk which is associated with random ergodic contraction operators on a Hilbert space, in analogy with the density of state functional for random self-adjoint operators. The DOS functional is shown to enjoy natural integral representations on the unit circle and on the unit disk. For random contractions with suitable finite volume approximations, the DOS functional is proven to be the almost sure infinite volume limit of the trace per unit volume of functions of the finite volume restrictions. Finally, in case the normalised counting measure of the spectrum of the finite volume restrictions converges in the infinite volume limit, the DOS functional is shown admit an integral representation on the disk in terms of the limiting measure, despite the discrepancy between the spectra of non normal operators and their finite volume restrictions. Moreover, the integral representation of the DOS functional on the unit circle is related to the Borel transform of the limiting measure.

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Spectral Properties of Non-Unitary Band Matrices

Cited by 2 publications

References 39 publications

Lower bounds on the localisation length of balanced random quantum walks

Lower bounds on the localisation length of balanced random quantum walks

Density of states for random contractions

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