The Lyapunov Exponent of Products of Random 2×2 Matrices Close to the Identity

Comtet, Alain; Luck, J. M.; Texier, Christophe; Tourigny, Yves

doi:10.1007/s10955-012-0674-8

Cited by 34 publications

(63 citation statements)

References 55 publications

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“…For the tight-binding problem, the localisation length is known to diverge as ξ ∼ 1/w 2 in the bulk of the spectrum, albeit only as ξ ∼ 1/w 2/3 in the vicinity of band edges [34,35,36,37,38]. Replacing ξ by the system size M and remembering that w stands for w/γ, we recover (72).…”

Section: Random Attractiveness Profilessupporting

confidence: 52%

On the coexistence of competing languages

Luck

Mehta

2020

Eur. Phys. J. B

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We investigate the evolution of competing languages, a subject where much previous literature suggests that the outcome is always the domination of one language over all the others. Since coexistence of languages is observed in reality, we here revisit the question of language competition, with an emphasis on uncovering the ways in which coexistence might emerge. We find that this emergence is related to symmetry breaking, and explore two particular scenarios -the first relating to an imbalance in the population dynamics of language speakers in a single geographical area, and the second to do with spatial heterogeneity, where language preferences are specific to different geographical regions. For each of these, the investigation of paradigmatic situations leads us to a quantitative understanding of the conditions leading to language coexistence. We also obtain predictions of the number of surviving languages as a function of various model parameters.

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Section: Random Attractiveness Profilessupporting

confidence: 52%

On the coexistence of competing languages

Luck

Mehta

2020

Eur. Phys. J. B

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“…This perturbative estimate breaks down near the band edges (E → ±2, i.e., q → 0 and π), where the numerator vanishes. Right at the band edges, eigenstates are actually more strongly localized, as their localization length only diverges as w −2/3 [32,33,34,35,36]. This anomalous band-edge scaling will play a key role in the following (see section 5).…”

Section: Particle In a Random Potential: Generalitiesmentioning

confidence: 90%

An investigation of equilibration in small quantum systems: the example of a particle in a 1D random potential

Luck¹

2016

J. Phys. A: Math. Theor.

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Abstract. We investigate the equilibration of a small isolated quantum system by means of its matrix of asymptotic transition probabilities in a preferential basis. The trace of this matrix is shown to measure the degree of equilibration of the system launched from a typical state, from the standpoint of the chosen basis. This approach is substantiated by an in-depth study of the example of a tightbinding particle in one dimension. In the regime of free ballistic propagation, the above trace saturates to a finite limit, testifying good equilibration. In the presence of a random potential, the trace grows linearly with the system size, testifying poor equilibration in the insulating regime induced by Anderson localization. In the weak-disorder situation of most interest, a universal finite-size scaling law describes the crossover between the ballistic and localized regimes. The associated crossover exponent 2/3 is dictated by the anomalous band-edge scaling characterizing the most localized energy eigenstates.

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“…As it is clear from its definition (75), the generalized Lyapunov exponent is the generating function of the cumulants of the logarithm of the wave function ln |y(x)| [more precisely, y(x) is the solution of the Cauchy initial value problem (71,70)]. As such, the GLE can be related to the large deviation function Φ(u) controlling the distribution Here we show how our method can be extended to study the mean number of equilibria at fixed values of the energy, defined, for our discrete model of an elastic line, as…”

Section: Large Deviations For the Wave Function Amplitudementioning

confidence: 99%

Exponential number of equilibria and depinning threshold for a directed polymer in a random potential

et al. 2018

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By extending the Kac-Rice approach to manifolds of finite internal dimension, we show that the mean number N tot of all possible equilibria (i.e. force-free configurations, a.k.a. equilibrium points) of an elastic line (directed polymer), confined in a harmonic well and submitted to a quenched random Gaussian potential in dimension d = 1 + 1, grows exponentially N tot ∼ exp (r L) with its length L. The growth rate r is found to be directly related to the generalized Lyapunov exponent (GLE) which is a momentgenerating function characterizing the large-deviation type fluctuations of the solution to the initial value problem associated with the random Schrödinger operator of the 1D Anderson localization problem. For strong confinement, the rate r is small and given by a non-perturbative (instanton, Lifshitz tail-like) contribution to GLE. For weak confinement, the rate r is found to be proportional to the inverse Larkin length of the pinning theory. As an application, identifying the depinning with a landscape "topology trivialization" phenomenon, we obtain an upper bound for the depinning threshold f c , in the presence of an applied force, for elastic lines and d-dimensional manifolds, expressed through the mean modulus of the spectral determinant of the Laplace operators with a random potential. We also discuss the question of counting of stable equilibria. Finally, we extend the method to calculate the asymptotic number of equilibria at fixed energy (elastic, potential and total), and obtain the (annealed) distribution of the energy density over these equilibria (i.e. force-free configurations). Some connections with the Larkin model are also established.

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The Lyapunov Exponent of Products of Random 2×2 Matrices Close to the Identity

Cited by 34 publications

References 55 publications

On the coexistence of competing languages

On the coexistence of competing languages

An investigation of equilibration in small quantum systems: the example of a particle in a 1D random potential

Exponential number of equilibria and depinning threshold for a directed polymer in a random potential

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