Study of the iterations of a mapping associated to a spin glass model

Collet, Pierre; Eckmann, Jean‐Pierre; Glaser, V.; Martin, A.

doi:10.1007/bf01224830

Cited by 26 publications

(69 citation statements)

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“…Let us mention that we also establish almost sure convergence of R n toward 1 when we are able to find γ ∈ (0, 1) and What one should expect at criticality is rather unclear to us (see however [23] for a number of predictions and numerical results on hierarchical pinning and also [6,7] for some theoretical considerations on a different class of hierarchical models).…”

Section: 5mentioning

confidence: 99%

“…Pinning models: the role of disorder. Hierarchical models on diamond lattices, homogeneous or disordered [5,4,6,7,9], are a powerful tool in the study of the critical behavior of statistical mechanics models, especially because real-space renormalization group transformationsà la Migdal-Kadanoff are exact in this case. In most of the cases, hierarchical models are introduced in association with a more realistic non-hierarchical one.…”

Section: 5mentioning

confidence: 99%

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Hierarchical pinning models, quadratic maps and quenched disorder

Giacomin

Lacoin

Toninelli

2009

Probab. Theory Relat. Fields

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Abstract. We consider a hierarchical model of polymer pinning in presence of quenched disorder, introduced by B. Derrida, V. Hakim and J. Vannimenus [11], which can be reinterpreted as an infinite dimensional dynamical system with random initial condition (the disorder). It is defined through a recurrence relation for the law of a random variable {Rn}n=1,2,..., which in absence of disorder (i.e., when the initial condition is degenerate) reduces to a particular case of the well-known Logistic Map. The large-n limit of the sequence of random variables 2 −n log Rn, a non-random quantity which is naturally interpreted as a free energy, plays a central role in our analysis. The model depends on a parameter α ∈ (0, 1), related to the geometry of the hierarchical lattice, and has a phase transition in the sense that the free energy is positive if the expectation of R0 is larger than a certain threshold value, and it is zero otherwise. It was conjectured in [11] that disorder is relevant (respectively, irrelevant or marginally relevant) if 1/2 < α < 1 (respectively, α < 1/2 or α = 1/2), in the sense that an arbitrarily small amount of randomness in the initial condition modifies the critical point with respect to that of the pure (i.e., non-disordered) model if α ≥ 1/2, but not if α < 1/2. Our main result is a proof of these conjectures for the case α = 1/2. We emphasize that for α > 1/2 we find the correct scaling form (for weak disorder) of the critical point shift.

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Section: 5mentioning

confidence: 99%

Section: 5mentioning

confidence: 99%

Hierarchical pinning models, quadratic maps and quenched disorder

Giacomin

Lacoin

Toninelli

2009

Probab. Theory Relat. Fields

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“…When the support of µ 0 is included in N, this question was previously answered by Collet et al [6]. By a careful study of the recursion equation for the generating functions of µ n , they proved that…”

Section: The Discrete-time Derrida-retaux Modelmentioning

confidence: 94%

“…where, for any x ∈ R, x + := max(x, 0), X n denotes an independent copy of X n , and (d) = stands for the identity in distribution. Note that this model was previously studied by Collet, Eckmann, Glaser and Martin [6] for random variables taking integer values. Through studying the sequence of probability-generating functions, they identified the critical manifold of (X n ), provided that X 0 ∈ N a.s. With renormalization group arguments, Derrida and Retaux studied this max-type recursive equation for real-valued random variables.…”

Section: Introductionmentioning

confidence: 99%

An Exactly Solvable Continuous-Time Derrida–Retaux Model

Mallein

Pain

2019

Commun. Math. Phys.

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To study the depinning transition in the limit of strong disorder, Derrida and Retaux [9] introduced a discrete-time max-type recursive model. It is believed that for a large class of recursive models, including Derrida and Retaux' model, there is a highly non-trivial phase transition. In this article, we present a continuous-time version of Derrida and Retaux model, built on a Yule tree, which yields an exactly solvable model belonging to this universality class. The integrability of this model allows us to study in details the phase transition near criticality and can be used to confirm the infinite order phase transition predicted by physicists. We also study the scaling limit of this model at criticality, which we believe to be universal. partially supported by ANR MALIN arXiv:1811.08749v2 [math.PR] 1 Apr 2019Note that the DR process can be seen as a discrete-time version of a solution of a McKean-Vlasov type SDE, see McKean [20], i.e. a Markov process interacting with its distribution. It follows from an easy recursion that for all n ∈ N, X n has law µ n .The aim of this article is to define continuous-time versions of the DR model, tree and process. In particular, we are looking for a process with a density r t (x) with respect to the Lebesgue measure which could solve [9, Equation (33)], that we recall here(1.8)This differential equation (1.8) plays a key role in the prediction (1.5), and was obtained by Derrida and Retaux as an informal scaling limit of the sequence of measures (µ nt (ndx)).

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“…(note that we do not require f to be normalized). This time evolution was introduced [11] to analyze a simple renormalization problem [7,8,11] which can be formulated as follows. Given a distribution P (X 0 ) of a positive random variable X 0 , what can be said on the distribution of the random variable X n constructed through the following recursion X n = max X (1) n−1 + X (2) n−1 − 1, 0…”

Section: Introductionmentioning

confidence: 99%

The critical behaviors and the scaling functions of a coalescence equation

Chen¹,

Dagard²,

Derrida³

et al. 2020

J. Phys. A: Math. Theor.

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We show that a coalescence equation exhibits a variety of critical behaviors, depending on the initial condition. This equation was introduced a few years ago to understand a toy model studied by Derrida and Retaux to mimic the depinning transition in presence of disorder. It was shown recently that this toy model exhibits the same critical behaviors as the equation studied in the present work. Here we find several families of exact solutions of this coalescence equation, in particular a family of scaling functions which are closely related to the different possible critical behaviors. These scaling functions lead to new conjectures, in particular on the shapes of the critical trees, that we have checked numerically.

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Study of the iterations of a mapping associated to a spin glass model

Cited by 26 publications

References 0 publications

Hierarchical pinning models, quadratic maps and quenched disorder

Hierarchical pinning models, quadratic maps and quenched disorder

An Exactly Solvable Continuous-Time Derrida–Retaux Model

The critical behaviors and the scaling functions of a coalescence equation

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