Most of works on 1D Anderson localisation consider the case where the potential has relatively small local fluctuations, such that <V<\/em>n<\/sub> 2<\/sup>> < \u221e (V<\/em>n<\/sub> is the on-site potenial for discrete models) or \u222b-\u221e<\/sub>+\u221e<\/sup>dx<\/em> <V<\/em>(x<\/em>)V<\/em>(0)> < \u221e (for continuous models). Models where this condition is not fulfilled lead to non standard localization properties (super-localisation) with non exponential damping of wave function’s envelope, like exp(-|x<\/em>|\u03b1<\/sup>) for \u03b1>1.<\/p>\n <\/p>\n We study the fluctuations<\/em> of the logarithm of the wave functions, a problem related to the analysis of the fluctuations of certain random matrix products:<\/p>\n This question is of importance in localisation problems : it is related to the discussion of the Single Parameter Scaling hypothesis ; this plays an important role when studying statistical properties of local density of states (Altshuler & Prigodin, 1989) or Wigner time delay (Texier & Comtet, Phys. Rev. Lett. 82<\/strong>(21), 4220 (1999)<\/a>).<\/p>\n See also<\/span> page \u201c products of random matrices<\/a> \u201d<\/strong><\/p>\n The \u00ab\u00a0single parameter scaling\u00a0\u00bb (SPS) hypothesis is a corner stone of the scaling theory of localization. It states that the full distribution of observable is controlled by a single characteristic scale (the localization length). First discussed within models with ad hoc random phase assumption (see the nice article: Cohen, Roth & Shapiro, Phys. Rev. B 38<\/strong>, 12125 (1988)<\/a>), the solvable Lloyd model has provided a ground to test SPS within a microscopic model (Deych, Lisyansky & Altshuler, Phys. Rev. Lett. 84<\/strong>, 2678 (2000)<\/a>). See also<\/span> page \u201c products of random matrices<\/a> \u201d<\/strong><\/p>\n Using this general formalism, I have provided in the previous paper a general framework allowing to analyse SPS in a very broad class of models with both finite of infinite second moment (like for the Lloyd model). A universal formula for the generalised Lyapunov exponent (cumulant generating function of ln|\u03c8(x)|) is derived:<\/p>\n Localization properties with potential with large local fluctuations Most of works on 1D Anderson localisation consider the case where the potential has relatively small local fluctuations, such that <Vn 2> < \u221e (Vn is the on-site potenial for discrete models) … Continuer la lecture \n
\nLocalization for one-dimensional random potentials with large local fluctuations<\/strong>
\nJ. Phys. A: Math. Theor. 41<\/strong>, 475001 (2008).<\/a> (9pp)
\ncond-mat arXiv:0807.0772.<\/a><\/li>\n<\/ul>\nFluctuations of random matrix products of SL(2,R) and localisation in the random mass Dirac equation<\/h3>\n
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\nFluctuations of random matrix products and 1D Dirac equation with random mass<\/strong>,
\n J. Stat. Phys. 157<\/strong>, 497-514 (2014)<\/a>
\ncond-mat arXiv:1402.6943<\/a><\/li>\n<\/ul>\nFluctuations and single parameter scaling for 1D disorder<\/h3>\n
\nIn the following paper, a general formula for the variance of ln|\u03c8(x)| is obtained, where \u03c8(x) solves the Schr\u00f6dinger equation, for arbitrary disorder characterised by its L\u00e9vy exponent L<\/em>(s<\/em>):<\/p>\n\n
\nFluctuations of the product of random matrices and generalized Lyapunov exponent<\/strong>,
\nJ. Stat. Phys (2020)<\/a>
\ncond-mat arXiv:1907.08512<\/a>
\nSome integral formula is derived for\u00a0\u03b32<\/sub>=limx<\/em>\u2192\u221e<\/sub>(1\/x)Var(ln|\u03c8(x)|) for the Schr\u00f6dinger equation with a random potential.<\/li>\n<\/ul>\n\n
\nGeneralized Lyapunov exponent of random matrices and universality classes for SPS in 1D Anderson localisation<\/strong>,
\nEurophys. Lett. 131, 17002 (2020)<\/a>
\ncond-mat arXiv:1910.01989<\/a>
\nUsing the general formalism of the previous article, the Single Parameter Scaling for Anderson localization is proven within a general framework, and extended to potentials with large fluctuations (such that <V2<\/sup>>=\u221e)<\/li>\n
\nThe generalized Lyapunov exponent for the one-dimensional Schr\u00f6dinger equation with Cauchy disorder: some exact results<\/strong>,
\ncond-mat arXiv:2110.01522<\/a>
\nTaking advantage of the specificity of the Lloyd model, we are able to get a secular equation for the generalized Lyapunov exponent, from which we derive a set of exact results<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"