The Effect of Disorder on Polymer Depinning Transitions

Alexander, Kenneth S.

doi:10.1007/s00220-008-0425-5

Cited by 74 publications

(350 citation statements)

References 21 publications

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“…n } i∈N are IID random variables and therefore this dynamical system is naturally re-interpreted as the evolution of the probability law L n (the law of R (1) n ): given L n , the law L n+1 is obtained by constructing two IID variables distributed according to L n and applying…”

mentioning

confidence: 99%

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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confidence: 99%

Hierarchical pinning models, quadratic maps and quenched disorder

Giacomin

Lacoin

Toninelli

2009

Probab. Theory Relat. Fields

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“…There exist several proofs of Theorem 2.1, with very different techniques: a second moment method in [1,39], martingale techniques in [34], and a large deviation/variational formula approach in [17]. The case α = 0 is studied in depth in [5]: the annealed and quenched critical points are equal, whatever β is.…”

Section: Results In the Iid Casementioning

confidence: 99%

“…In his seminal paper, Harris [32] explains that disorder relevance can be decided only through the order ν hom of the homogeneous phase transition: the correlation length of the homogeneous system diverges as |h − h c | −ν hom +o (1) as h approaches h c (a precise definition of the order of the phase transition is given in the context of pinning model, see (5)). Harris predicts that an IID disorder is relevant if ν hom < 2/d and irrelevant if ν hom > 2/d.…”

Section: The Harris Criterion For General Disordered Systemsmentioning

confidence: 99%

Influence of Disorder For the Polymer Pinning Model

Berger

2015

ESAIM: Proc.

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Abstract. When studying physical systems, the influence of disorder on the phase transition is a central question: one wants to determine whether an arbitrary small amount of randomness modifies the critical properties of the system, with respect to the non-disordered case. We present here an overview of the mathematical results obtained to answer that question in the context of the polymer pinning model. In the IID case, the picture of disorder relevance/irrelevance is by now established, and follows the so-called Harris criterion: disorder is irrelevant if ν hom > 2 and relevant if ν hom < 2, where ν hom is the order of the homogeneous phase transition. The marginal case ν hom = 2 has been subject to controversy in the physics literature in the context of pinning models, but has recently been fully settled. In the correlated case, Weinrib and Halperin predicted that, if the two-point correlations decay as a power law with exponent ξ > 0, then the Harris criterion would be modified if ξ < 1: disorder should be relevant whenever ν hom < 2 max(1, 1/ξ). It turns out that this prediction is not accurate: the key quantity is not the decay exponent ξ, but the occurrence of rare regions with atypical disorder. An infinite disorder regime may appear, in which the relevance/irrelevance picture is crucially modified. We also mention another recent approach to the question of the influence of disorder for the pinning model: the persistence of disorder when taking the scaling limit of the system.

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“…As we have pointed out, this was to be expected: can one still extract from (4.2) some interesting information? The answer is positive, as shown by K. Alexander in [4]. The crucial point is not to take the limit in N , but rather exploit (4.2) up to the scale of the correlation length of the annealed system, which is just a homogeneous system with pinning potential δ (see Remark 2.4).…”

Section: Free Energy Lower Bounds and Irrelevant Disorder Estimatesmentioning

confidence: 98%

“…The results in Theorem 3.6 are almost sharp, because in [4,54] it is proven that for every K(·) such that α ∈ (1/2, 1) there exists C > 0 such that…”

Section: 4mentioning

confidence: 99%

Renewal Sequences, Disordered Potentials, and Pinning Phenomena

Giacomin

2009

Spin Glasses: Statics and Dynamics

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Abstract. We give an overview of the state of the art of the analysis of disordered models of pinning on a defect line. This class of models includes a number of well known and much studied systems (like polymer pinning on a defect line, wetting of interfaces on a disordered substrate and the Poland-Scheraga model of DNA denaturation). A remarkable aspect is that, in absence of disorder, all the models in this class are exactly solvable and they display a localization-delocalization transition that one understands in full detail. Moreover the behavior of such systems near criticality is controlled by a parameter and one observes, by tuning the parameter, the full spectrum of critical behaviors, ranging from first order to infinite order transitions. This is therefore an ideal set-up in which to address the question of the effect of disorder on the phase transition, notably on critical properties. We will review recent results that show that the physical prediction that goes under the name of Harris criterion is indeed fully correct for pinning models. Beyond summarizing the results, we will sketch most of the arguments of proof.2000 Mathematics Subject Classification: 82B44, 60K37, 60K35

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The Effect of Disorder on Polymer Depinning Transitions

Cited by 74 publications

References 21 publications

Hierarchical pinning models, quadratic maps and quenched disorder

Hierarchical pinning models, quadratic maps and quenched disorder

Influence of Disorder For the Polymer Pinning Model

Renewal Sequences, Disordered Potentials, and Pinning Phenomena

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