A Numerical Approach to Copolymers at Selective Interfaces

Caravenna, Francesco; Giacomin, Giambattista; Gubinelli, Massimiliano

doi:10.1007/s10955-005-8081-z

Cited by 23 publications

(36 citation statements)

References 24 publications

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“…The case (λ, h) ∈ L 1 is harder to investigate and we must recall that the strict inequality h(λ) < h 0 c (β) is not rigorously proven for the moment. However, some numerical evidences in [7] shows that L 1 is not an empty set and contrary to what we just said about D and L 2 , the influence of a depinning term in the region L 1 is not understood at all. This leads to the following open problem: for (λ, h) ∈ L 1 , namely when h(λ) ≤ h < h 0 c (λ), can we find a large enough depinning term β < 0 that leads to a delocalization, i.e., h ≥ h β c (λ)?…”

Section: Two Particular Casescontrasting

confidence: 69%

Copolymer at selective interfaces and pinning potentials: Weak coupling limits

Pétrélis¹

2009

Ann. Inst. H. Poincaré Probab. Statist.

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We consider a simple random walk of length N , denoted by (Si) i∈{1,...,N} , and we define (wi) i≥1 a sequence of centered i.i.d. random variables. For K ∈ N we define ((γ −K i , . . . , γ K i )) i≥1 an i.i.d sequence of random vectors. We set β ∈ R, λ ≥ 0 and h ≥ 0, and transform the measure on the set of random walk trajectories with the HamiltonianThis transformed path measure describes an hydrophobic(philic) copolymer interacting with a layer of width 2K around an interface between oil and water.In the present article we prove the convergence in the limit of weak coupling (when λ, h and β tend to 0) of this discrete model towards its continuous counterpart. To that aim we further develop a technique of coarse graining introduced by Bolthausen and den Hollander in [6]. Our result shows, in particular, that the randomness of the pinning around the interface vanishes as the coupling becomes weaker.

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Section: Two Particular Casescontrasting

confidence: 69%

Copolymer at selective interfaces and pinning potentials: Weak coupling limits

Pétrélis¹

2009

Ann. Inst. H. Poincaré Probab. Statist.

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“…It is also known (see [3] and [17], Chapter 5.2) that h c (β) < h c (0) for every β > 0. 9 The annealed bound (2.6) applied to this case shows that…”

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confidence: 90%

Disordered pinning models and copolymers: Beyond annealed bounds

Toninelli¹

2008

Ann. Appl. Probab.

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We consider a general model of a disordered copolymer with adsorption. This includes, as particular cases, a generalization of the copolymer at a selective interface introduced by Garel et al. [Europhys. Lett. 8 (1989) 9-13], pinning and wetting models in various dimensions, and the Poland-Scheraga model of DNA denaturation. We prove a new variational upper bound for the free energy via an estimation of noninteger moments of the partition function. As an application, we show that for strong disorder the quenched critical point differs from the annealed one, for example, if the disorder distribution is Gaussian. In particular, for pinning models with loop exponent 0 < α < 1/2 this implies the existence of a transition from weak to strong disorder. For the copolymer model, under a (restrictive) condition on the law of the underlying renewal, we show that the critical point coincides with the one predicted via renormalization group arguments in the theoretical physics literature. A stronger result holds for a "reduced wetting model" introduced by Bodineau and Giacomin [J. Statist. Phys. 117 (2004) 801-818]: without restrictions on the law of the underlying renewal, the critical point coincides with the corresponding renormalization group prediction. . This reprint differs from the original in pagination and typographic detail. 1 2 F. L. TONINELLI present a nontrivial localization-delocalization phase transition due to an energy-entropy competition.Mathematically, the model is defined in terms of a renewal sequence whose inter-arrival law has a power-like tail with exponent α + 1 ≥ 1. The model is exactly solvable in absence of disorder, and it turns out that the transition can be of any given order, from first to infinite, according to the value of α. This is therefore an ideal testing ground for physical arguments (Harris criterion, renormalization group computations) and predictions concerning the effect of disorder on the critical exponents and on the location of the critical point.The comprehension of this model has witnessed remarkable progress on the mathematical side, as proved by the recent book [17]. In particular it has been shown that for wetting, pinning or PS models, an arbitrary amount of disorder modifies the free-energy critical exponent if α > 1/2 [18], that is, disorder is relevant in this case, in agreement with the predictions of the so-called Harris criterion [23]. On the other hand, for 0 < α < 1/2 it has been proven recently [2,29] that if disorder is weak enough the free-energy critical exponent coincides with that of the homogeneous (i.e., nondisordered) model, and the (quenched) critical point coincides with the annealed one: disorder is irrelevant (again, in agreement with the Harris criterion). These results about "irrelevance" of disorder for 0 < α < 1/2 have been later refined and complemented in [21] with results about correlation-length critical exponents. The marginal case α = 1/2 is strongly debated in the theoretical physics literature: Ref. [14] claims that quenched and annealed c...

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“…Here physicists generally take "altered" to mean that the specific heat exponent is different, but a disorder-induced change in the critical point is also of interest ( [4], [5], [14], [17]). This produces the question, "change relative to what, u a c or u q x ?"…”

Section: Introductionmentioning

confidence: 99%

The Effect of Disorder on Polymer Depinning Transitions

Alexander

2008

Commun. Math. Phys.

339

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Abstract. We consider a polymer, with monomer locations modeled by the trajectory of a Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. We assume that probability of an excursion of length n is given by n −c ϕ(n) for some 1 < c < 2 and slowly varying ϕ. Disorder is introduced by having the interaction vary from one monomer to another, as a constant u plus i.i.d. mean-0 randomness. There is a critical value of u above which the polymer is pinned, placing a positive fraction (called the contact fraction) of its monomers at 0 with high probability. To see the effect of disorder on the depinning transition, we compare the contact fraction and free energy (as functions of u) to the corresponding annealed system. We show that for c > 3/2, at high temperature, the quenched and annealed curves differ significantly only in a very small neighborhood of the critical point-the size of this neighborhood scales as β 1/(2c−3) where β is the inverse temperature. For c < 3/2, given ǫ > 0, for sufficiently high temperature the quenched and annealed curves are within a factor of 1 − ǫ for all u near the critical point; in particular the quenched and annealed critical points are equal. For c = 3/2 the regime depends on the slowly varying function ϕ.

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A Numerical Approach to Copolymers at Selective Interfaces

Cited by 23 publications

References 24 publications

Copolymer at selective interfaces and pinning potentials: Weak coupling limits

Copolymer at selective interfaces and pinning potentials: Weak coupling limits

Disordered pinning models and copolymers: Beyond annealed bounds

The Effect of Disorder on Polymer Depinning Transitions

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