Products of Random Matrices and Generalised Quantum Point Scatterers

Comtet, Alain; Texier, Christophe; Tourigny, Yves

doi:10.1007/s10955-010-0005-x

Cited by 27 publications

(64 citation statements)

References 41 publications

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“…It is striking that Formula (11.4) for γ µ looks much more complicated than the formula for γ(λ + i0). The fact the both formulae are correct is proved in Comtet et al (2010). We shall not reproduce the proof here, but some of the calculations in later sections will illustrate the simplifications that can occur when applying Furstenberg's formula to specific products of 2 × 2 matrices.…”

Section: Notationmentioning

confidence: 96%

“…Recommended reading. The lectures constitute an extensive development of some of the ideas presented in the papers by Comtet, Texier and Tourigny (2010;2011; and Comtet, Luck, Texier and Tourigny (2013); from the mathematical point of view, they are more or less self-contained, in the sense that anyone who is familiar with the basic concepts of probability, differential equations, complex variables, and group theory should be able to learn something from them.…”

Section: 3mentioning

confidence: 99%

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Stochastic Processes and Random Matrices

2017

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-terms arXiv:1601.01822v1 [math-ph] 8 Jan 2016 IMPURITY MODELS AND PRODUCTS OF RANDOM MATRICES ALAIN COMTET AND YVES TOURIGNYAbstract. This is an extended version of lectures given at the Summer School on Stochastic Processes and Random Matrices, held at theÉcole de Physique, Les Houches, in July 2015. The aim is to introduce the reader to the theory of one-dimensional disordered systems and products of random matrices, confined to the 2×2 case. The notion of impurity model-that is, a system in which the interactions are highly localised-links the two themes and enables their study by elementary mathematical tools. After discussing the spectral theory of some impurity models, we state and illustrate Furstenberg's theorem, which gives sufficient conditions for the exponential growth of a product of independent, identically-distributed matrices.

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Section: Notationmentioning

confidence: 96%

Section: 3mentioning

confidence: 99%

Stochastic Processes and Random Matrices

2017

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“…The mapping between RMP and 1D localisation models like the random Kronig-Penney model [3], was recently extended to general RMPs of SL(2, R) [16]. For the case of interest here, the mapping works as follows : consider a random mass given as a superposition of delta-functions m(x) = n η n δ(x − x n ), where coordinates are ordered x 1 < x 2 < · · · .…”

Section: Mappingmentioning

confidence: 99%

Fluctuations of Random Matrix Products and 1D Dirac Equation with Random Mass

Ramola

Texier

2014

J Stat Phys

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We study the fluctuations of certain random matrix products, describing localisation properties of the one-dimensional Dirac equation with random mass. In the continuum limit, i.e. when matrices M n 's are close to the identity matrix, we obtain convenient integral representations for the variance Γ 2 = lim N →∞ Var(ln ||Π N ||)/N . The case studied exhibits a saturation of the variance at low energy ε along with a vanishing Lyapunov exponent Γ 1 = lim N →∞ ln ||Π N ||/N , leading to the behaviour Γ 2 /Γ 1 ∼ ln(1/|ε|) → ∞ as ε → 0. Our continuum description sheds new light on the Kappus-Wegner (band center) anomaly.

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“…It generally requires solving an integral equation to obtain the invariant measure of a continuous diffusion process on a real projective space [21]. In low dimensions this can be done with numerical quadrature [4], but this is not tractable for our high dimensional problem.…”

Section: Computing the Error Exponentmentioning

confidence: 99%

Detecting randomwalks hidden in noise: Phase transition on large graphs

Agaskar

2013

2013 IEEE International Conference on Acoustics, Speech and Signal Processing

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We consider the problem of distinguishing between two hypotheses: that a sequence of signals on a large graph consists entirely of noise, or that it contains a realization of a random walk buried in the noise. The problem of computing the error exponent of the optimal detector is simple to formulate, but exhibits deep connections to problems known to be difficult, such as computing Lyapunov exponents of products of random matrices and the free entropy density of statistical mechanical systems. We describe these connections, and define an algorithm that efficiently computes the error exponent of the Neyman-Pearson detector. We also derive a closed-form formula, derived from a statistical mechanics-based approximation, for the error exponent on an arbitrary graph of large size. The derivation of this formula is not entirely rigorous, but it closely matches the empirical results in all our experiments. This formula explains a phase transition phenomenon in the error exponent: below a threshold SNR, the error exponent is nearly constant and near zero, indicating poor performance; above the threshold, there is rapid improvement in performance as the SNR increases. The location of the phase transition depends on the entropy rate of the random walk.

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Products of Random Matrices and Generalised Quantum Point Scatterers

Cited by 27 publications

References 41 publications

Stochastic Processes and Random Matrices

Stochastic Processes and Random Matrices

Fluctuations of Random Matrix Products and 1D Dirac Equation with Random Mass

Detecting randomwalks hidden in noise: Phase transition on large graphs

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