Regular Expansion for the Characteristic Exponent of a Product of 2 × 2 Random Matrices

Havret, Benjamin

doi:10.1007/s11040-019-9312-x

Cited by 4 publications

(5 citation statements)

References 25 publications

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“…Results in this direction can be found in [15] that we cite also for more complete results. Let us point out, however, that identifying the singular behavior, that is the presence of C α ε 2|α| , is an open problem and the approach we employ does not seem to be appropriate when this is a subleading term (i.e., |α| ≥ 1).…”

Section: 3mentioning

confidence: 82%

Lyapunov exponent for products of random Ising transfer matrices: the balanced disorder case

Giacomin,

Greenblatt

2021

Preprint

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We analyze the top Lyapunov exponent of the transfer matrix for the nearest neighbor Ising chain with random external field. This Lyapunov exponent coincides with the free energy of the Ising chain with random external field, but it also plays a central role in the analysis of the two dimensional Ising model with columnar disorder and of the quantum chain with transverse random field. We obtain its sharp behavior in the large interaction limit when the external field is centered: this balanced case turns out to be critical in many respects. From a mathematical standpoint we precisely identify the behavior of the top Lyapunov exponent of a product of two dimensional random matrices close to a diagonal random matrix for which top and bottom Lyapunov exponents coincide. In particular, the Lyapunov exponent is only log-Hölder continuous.

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Section: 3mentioning

confidence: 82%

Lyapunov exponent for products of random Ising transfer matrices: the balanced disorder case

Giacomin,

Greenblatt

2021

Preprint

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“…Let us also point out that exponents ν defined in a similar manner to (4) played a role in [5,8,10]. These papers looked at the Lyapunov exponent near a critical value (corresponding to a critical energy in our terminology described below) and exhibited singular behavior of the Lyapunov exponent in its vicinity, namely a deviation from the standard quadratic vanishing of the Lyapunov exponent.…”

Section: Intuition and Main Resultsmentioning

confidence: 99%

Pseudo-gaps for random hopping models

Dorsch

Schulz-Baldes

2020

J. Phys. A: Math. Theor.

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For one-dimensional random Schrödinger operators, the integrated density of states is known to be given in terms of the (averaged) rotation number of the Prüfer phase dynamics. This paper develops a controlled perturbation theory for the rotation number around an energy, at which all the transfer matrices commute and are hyperbolic. Such a hyperbolic critical energy appears in random hopping models. The main result is a Hölder continuity of the rotation number at the critical energy that, under certain conditions on the randomness, implies the existence of a pseudo-gap. The proof uses renewal theory. The result is illustrated by numerics.

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“…A result about (1.26), i.e. for |α| ě 1, has been achieved recently [24], but the the expansion is fully controlled only up to (and excluding) the singular term Cε 2α : for the moment results about this term remain very weak.…”

Section: 2mentioning

confidence: 94%

“…for matrix products (without assuming the disorder to be small). In particular, the ε 2α singularity found in [12] is fully present in the continuum limit expression: mathematical results on this issue for p L Z ∆ pεq have been recently obtained [20,24], but the control of the singular term is an open problem for |α| ě 1.…”

Section: Introductionmentioning

confidence: 99%

Continuum Limit of Random Matrix Products in Statistical Mechanics of Disordered Systems

Comets

Giacomin

Greenblatt

2019

Commun. Math. Phys.

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We consider a particular weak disorder limit (continuum limit) of matrix products that arise in the analysis of disordered statistical mechanics systems, with a particular focus on random transfer matrices. The limit system is a diffusion model for which the leading Lyapunov exponent can be expressed explicitly in terms of modified Bessel functions, a formula that appears in the physical literature on these disordered systems. We provide an analysis of the diffusion system as well as of the link with the matrix products. We then apply the results to the framework considered by Derrida and Hilhorst in [12], which deals in particular with the strong interaction limit for disordered Ising model in one dimension and that identifies a singular behavior of the Lyapunov exponent (of the transfer matrix), and to the two dimensional Ising model with columnar disorder (McCoy-Wu model). We show that the continuum limit sharply captures the Derrida and Hilhorst singularity. Moreover we revisit the analysis by McCoy and Wu [31] and remark that it can be interpreted in terms of the continuum limit approximation. We provide a mathematical analysis of the continuum approximation of the free energy of the McCoy-Wu model, clarifying the prediction (by McCoy and Wu) that, in this approximation, the free energy of the two dimensional Ising model with columnar disorder is C 8 but not analytic at the critical temperature.

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Regular Expansion for the Characteristic Exponent of a Product of 2 × 2 Random Matrices

Cited by 4 publications

References 25 publications

Lyapunov exponent for products of random Ising transfer matrices: the balanced disorder case

Lyapunov exponent for products of random Ising transfer matrices: the balanced disorder case

Pseudo-gaps for random hopping models

Continuum Limit of Random Matrix Products in Statistical Mechanics of Disordered Systems

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