On the Delocalized Phase of the Random Pinning Model

Mourrat, Jean-Christophe

doi:10.1007/978-3-642-27461-9_18

Cited by 4 publications

(3 citation statements)

References 9 publications

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“…However, no formulas were obtained for the relevant constants.The latter two papers prove the bound under the average quenched measure, i.e., under E(P β,h,ω n ). For the pinning model with disorder, the same result as in Corollary 1.6 was derived in Mourrat [24] with the help of the variational characterization obtained in Cheliotis and den Hollander [13].…”

mentioning

confidence: 61%

A copolymer near a selective interface: Variational characterization of the free energy

Bolthausen¹,

Hollander²,

Opoku³

2015

Ann. Probab.

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In this paper we consider a random copolymer near a selective interface separating two solvents. The configurations of the copolymer are directed paths that can make i.i.d. excursions of finite length above and below the interface. The excursion length distribution is assumed to have a tail that is logarithmically equivalent to a power law with exponent α ≥ 1. The monomers carry i.i.d. real-valued types whose distribution is assumed to have zero mean, unit variance, and a finite moment generating function. The interaction Hamiltonian rewards matches and penalizes mismatches of the monomer types and the solvents, and depends on two parameters: the interaction strength β ≥ 0 and the interaction bias h ≥ 0. We are interested in the behavior of the copolymer in the limit as its length tends to infinity.The quenched free energy per monomer (β, h) → g que (β, h) has a phase transition along a quenched critical curve β → h que c (β) separating a localized phase, where the copolymer stays close to the interface, from a delocalized phase, where the copolymer wanders away from the interface. We derive variational formulas for both these quantities. We compare these variational formulas with their analogues for the annealed free energy per monomer (β, h) → g ann (β, h) and the annealed critical curve β → h ann c (β

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confidence: 61%

A copolymer near a selective interface: Variational characterization of the free energy

Bolthausen¹,

Hollander²,

Opoku³

2015

Ann. Probab.

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“…7], whereas in the disordered case the situation is more subtle, since there might be some stretches of unusual disorder driving the polymer to have a growing number of contacts, even if h < h que c (β). One a priori only knows that the number of contacts is o(N ) in the delocalized regime, but [35] gives that the number of contact under P β,ω N,h is O(log N ) P-a.s. In [6], the authors show that the last contact occurs at distance O(1) of the origin in probability (cf.…”

Section: Remark 23 the Condition N∈nmentioning

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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“…In proving Theorem 1.1 we will make use of the following theorem, which was proved in [14]. 14]). Let β ≥ 0 and h < h c (β).…”

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confidence: 99%

Path properties of the disordered pinning model in the delocalized regime

Alexander¹,

Zygouras²

2014

Ann. Appl. Probab.

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We study the path properties of a random polymer attracted to a defect line by a potential with disorder, and we prove that in the delocalized regime, at any temperature, the number of contacts with the defect line remains in a certain sense "tight in probability" as the polymer length varies. On the other hand we show that at sufficiently low temperature, there exists a.s. a subsequence where the number of contacts grows like the log of the length of the polymer.

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On the Delocalized Phase of the Random Pinning Model

Cited by 4 publications

References 9 publications

A copolymer near a selective interface: Variational characterization of the free energy

A copolymer near a selective interface: Variational characterization of the free energy

Influence of Disorder For the Polymer Pinning Model

Path properties of the disordered pinning model in the delocalized regime

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