Fractional Moment Bounds and Disorder Relevance for Pinning Models

Derrida, Bernard; Giacomin, Giambattista; Lacoin, Hubert; Toninelli, Fabio Lucio

doi:10.1007/s00220-009-0737-0

Cited by 79 publications

(189 citation statements)

References 33 publications

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“…Theorem 2.2 collects several results proven in a series of papers [4,10,16,17,19,28,29,31] in several manners, the full necessary and sufficient condition for disorder relevance (together with the sharp critical point shift when α = 1/2) being given only recently in [12]. In [12,19,28,29], the authors estimate the fractional moment of the partition function up to the correlation length by a change of measure argument, and then use a coarse-graining procedure to glue these estimates together.…”

Section: Results In the Iid Casementioning

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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Section: Results In the Iid Casementioning

confidence: 99%

Influence of Disorder For the Polymer Pinning Model

Berger

2015

ESAIM: Proc.

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“…it has been shown in [7] that in (3.27) one can take ε = 0 and c 0 is still positive. Theorem 3.6 has been proven in [19] and we give an outline of the proof in Section 5. The method is based on estimating fractional moments of the free energy, while (3.29) is derived by adapting the techniques yielding Theorem 3.5 (and a sketch of the proof is in Section 4).…”

Section: 4mentioning

confidence: 99%

“…A (rough) bound in the other direction is obtained by neglecting P h (n ∈ τ )K(N − n) in the right-hand side of (2.17), so that 19) which holds once again for every N . The sharp asymptotic result is…”

Section: The Critical Behavior Of the Free Energy Is Given Bymentioning

confidence: 99%

“…The method is based on estimating fractional moments of the free energy, while (3.29) is derived by adapting the techniques yielding Theorem 3.5 (and a sketch of the proof is in Section 4). In [19] the case α = 1 is not considered, but in [10] it is shown that the fractional moment method can be generalized to establish in particular that h c (β) > h ann c (β) also for α = 1.…”

Section: 4mentioning

confidence: 99%

“…To go beyond the estimate we have just presented, in [19] another renewal identity has been exploited, namely: for every fixed k and every…”

Section: Iterated Fractional Moment Estimatesmentioning

confidence: 99%

See 2 more Smart Citations

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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Marginal relevance of disorder for pinning models

Giacomin

Lacoin

Toninelli

2009

Comm Pure Appl Math

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134

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Abstract. The effect of disorder on pinning and wetting models has attracted much attention in theoretical physics (e.g. [16,11]). In particular, it has been predicted on the basis of the Harris criterion that disorder is relevant (annealed and quenched model have different critical points and critical exponents) if the return probability exponent α, a positive number that characterizes the model, is larger than 1/2. Weak disorder has been predicted to be irrelevant (i.e. coinciding critical points and exponents) if α < 1/2. Recent mathematical work (in particular [2,10,21,22]) has put these predictions on firm grounds. In renormalization group terms, the case α = 1/2 is a marginal case and there is no agreement in the literature as to whether one should expect disorder relevance [11] or irrelevance [16] at marginality. The question is particularly intriguing also because the case α = 1/2 includes the classical models of two-dimensional wetting of a rough substrate, of pinning of directed polymers on a defect line in dimension (3 + 1) or (1 + 1) and of pinning of an heteropolymer by a point potential in three-dimensional space. Here we prove disorder relevance both for the general α = 1/2 pinning model and for the hierarchical version of the model proposed in [11], in the sense that we prove a shift of the quenched critical point with respect to the annealed one. In both cases we work with Gaussian disorder and we show that the shift is at least of order exp(−1/β 4 ) for β small, if β 2 is the disorder variance.2000 Mathematics Subject Classification: 82B44, 60K37, 60K05, 82B41

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Fractional Moment Bounds and Disorder Relevance for Pinning Models

Cited by 79 publications

References 33 publications

Influence of Disorder For the Polymer Pinning Model

Influence of Disorder For the Polymer Pinning Model

Renewal Sequences, Disordered Potentials, and Pinning Phenomena

Marginal relevance of disorder for pinning models

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