Publication
Lyapunov exponents of Green's functions for random potentials tending to zero
2011
Journal paper
Journal paper
Abstract
We consider quenched and annealed Lyapunov exponents for the Green's function of -Delta + gamma V, where the potentials V(x) ,x epsilon 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 root 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 Wang. We also consider other ways to send the potential to zero than multiplying it by a small number.
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Related concepts (24)
Lyapunov exponent
In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation vector diverge (provided that the divergence can be treated within the linearized approximation) at a rate given by where is the Lyapunov exponent. The rate of separation can be different for different orientations of initial separation vector.
Evoked potential
An evoked potential or evoked response is an electrical potential in a specific pattern recorded from a specific part of the nervous system, especially the brain, of a human or other animals following presentation of a stimulus such as a light flash or a pure tone. Different types of potentials result from stimuli of different modalities and types. Evoked potential is distinct from spontaneous potentials as detected by electroencephalography (EEG), electromyography (EMG), or other electrophysiologic recording method.
Event-related potential
An event-related potential (ERP) is the measured brain response that is the direct result of a specific sensory, cognitive, or motor event. More formally, it is any stereotyped electrophysiological response to a stimulus. The study of the brain in this way provides a noninvasive means of evaluating brain functioning. ERPs are measured by means of electroencephalography (EEG). The magnetoencephalography (MEG) equivalent of ERP is the ERF, or event-related field. Evoked potentials and induced potentials are subtypes of ERPs.
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