Lyapunov exponents of Green's functions for random potentials tending to zero

We consider quenched and annealed Lyapunov exponents for the Green's function of \(-\Delta+\gamma V\), where the potentials \(V(x), x\in\Z^d\), are i.i.d. nonnegative random variables and \(\gamma>0\) is a scalar. We present a probabilistic proof that both Lyapunov exponents scale like \(c\s...

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Bibliographic Details
Published in:arXiv.org 2010-03
Main Authors: Kosygina, Elena, Mountford, Thomas S, Zerner, Martin P W
Format: Article
Language:English
Subjects:
Annealing
Chaos theory
Green's functions
Liapunov exponents
Quenching
Random variables
Online Access:Get full text
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Summary:We consider quenched and annealed Lyapunov exponents for the Green's function of \(-\Delta+\gamma V\), where the potentials \(V(x), x\in\Z^d\), are i.i.d. nonnegative random variables and \(\gamma>0\) is a scalar. We present a probabilistic proof that both Lyapunov exponents scale like \(c\sqrt{\gamma}\) as \(\gamma\) tends to 0. Here the constant \(c\) is the same for the quenched as for the annealed exponent and is computed explicitly. This improves results obtained previously by Wei-Min Wang. We also consider other ways to send the potential to zero than multiplying it by a small number.
ISSN:2331-8422
DOI:10.48550/arxiv.0903.4928