On the distribution of a random variable occurring in 1D disordered systems

Calan, C de; Luck, J. M.; Nieuwenhuizen, Th. M.; Petritis, Dimitri

doi:10.1088/0305-4470/18/3/025

Cited by 108 publications

(105 citation statements)

References 23 publications

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“…the asymptotic form of the scaling functionp for the canonical ensemple was determined analytically [2], for particular distributions of the fields and/or couplings it can even be calculated exactly [21]. The average is determined by the rare events having a magnetization of order O(1), i.e.…”

Section: Critical Properties a Surface Magnetization -Canonical Vmentioning

confidence: 99%

Random transverse Ising spin chain and random walks

1998

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We study the critical and off-critical (Griffiths-McCoy) regions of the random transverse-field Ising spin chain by analytical and numerical methods and by phenomenological scaling considerations. Here we extend previous investigations to surface quantities and to the ferromagnetic phase. The surface magnetization of the model is shown to be related to the surviving probability of an adsorbing walk and several critical exponents are exactly calculated. Analyzing the structure of low energy excitations we present a phenomenological theory which explains both the scaling behavior at the critical point and the nature of Griffiths-McCoy singularities in the off-critical regions. In the numerical part of the work we used the free-fermion representation of the model and calculated the critical magnetization profiles, which are found to follow very accurately the conformal predictions for different boundary conditions. In the off-critical regions we demonstrated that the GriffithsMcCoy singularities are characterized by a single, varying exponent, the value of which is related through duality in the paramagnetic and ferromagnetic phases.

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Section: Critical Properties a Surface Magnetization -Canonical Vmentioning

confidence: 99%

Random transverse Ising spin chain and random walks

1998

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“…In some cases, when the scale invariance is discrete [16,36,14,32,29,18,2,1], the amplitude of the powerlaw acquires a periodicity, often called log-periodic oscillations: see [35,21] for reviews on the topic. These oscillatory amplitudes are usually more difficult to calculate [13,12,7,26] than the critical exponents. When the scale invariance is associated to a renormalization group map, they can be related to some properties of the map, for example to the shape of its Julia set, see [13,8].…”

Section: Introductionmentioning

confidence: 99%

Log-periodic Critical Amplitudes: A Perturbative Approach

Derrida

Giacomin

2013

J Stat Phys

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Abstract. Log-periodic amplitudes appear in the critical behavior of a large class of systems, in particular when a discrete scale invariance is present. Here we show how to compute these critical amplitudes perturbatively when they originate from a renormalization map which is close to a monomial. In this case, the log-periodic amplitudes of the subdominant corrections to the leading critical behavior can also be calculated.Dedicated to Herbert Spohn, our colleague, friend and source of inspiration, on the occasion of his 65 th birthday

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“…The proof is based on the characterization of the domain of analyticity of the Mellin transforms of G ν 0 and G ω 0 . As shown in [6], the poles of the Mellin transform of G ν 0 (s) are either roots of (1.10) or at integer translates of those roots (the latter are not important for our result); the relevant argument is summarized in Section 4.1 for completeness. We are not able to control the behavior of the Mellin transform well enough to use it to directly obtain an asymptotic expression for G ν 0 with control on the remainder, but we are instead able to do this (in Section 4.2) at the level of the primitive of G ν 0 and then then recover the desired result on G ν 0 by reinjecting the estimate into the fixed point equation satisfied by G ν 0 .…”

Section: L(ε) = Limmentioning

confidence: 89%

Singular Behavior of the Leading Lyapunov Exponent of a Product of Random $${2 \times 2}$$ 2 × 2 Matrices

Genovese

Giacomin

Greenblatt

2017

Commun. Math. Phys.

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Abstract. We consider a certain infinite product of random 2×2 matrices appearing in the solution of some 1 and 1 + 1 dimensional disordered models in statistical mechanics, which depends on a parameter ε > 0 and on a real random variable with distribution µ. For a large class of µ, we prove the prediction by B. Derrida and H. J. Hilhorst (J. Phys. A 16, 1641Phys. A 16, -2654Phys. A 16, (1983) that the Lyapunov exponent behaves like Cε 2α in the limit ε 0, where α ∈ (0, 1) and C > 0 are determined by µ. Derrida and Hilhorst performed a two-scale analysis of the integral equation for the invariant distribution of the Markov chain associated to the matrix product and obtained a probability measure that is expected to be close to the invariant one for small ε. We introduce suitable norms and exploit contractivity properties to show that such a probability measure is indeed close to the invariant one in a sense which implies a suitable control of the Lyapunov exponent.

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On the distribution of a random variable occurring in 1D disordered systems

Cited by 108 publications

References 23 publications

Random transverse Ising spin chain and random walks

Random transverse Ising spin chain and random walks

Log-periodic Critical Amplitudes: A Perturbative Approach

Singular Behavior of the Leading Lyapunov Exponent of a Product of Random $${2 \times 2}$$ 2 × 2 Matrices

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