Copolymer at selective interfaces and pinning potentials: Weak coupling limits

Pétrélis, Nicolas

doi:10.1214/07-aihp160

Cited by 5 publications

(5 citation statements)

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“…In fact, Theorem 1.3 has been generalized in [17] to a large class of disorder random variables (including all bounded random variables). A further generalization has been obtained in [21], in the case when, added to the copolymer interaction, there is also a pinning interaction at the interface, that is, an energy reward in touching the interface. We stress, however, that these generalizations are always for the copolymer model built over the simple random walk: going beyond the simple random walk case appears indeed to be a very delicate (albeit natural) step.…”

Section: Introductionmentioning

confidence: 94%

The weak coupling limit of disordered copolymer models

Caravenna¹,

Giacomin²

2010

Ann. Probab.

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A copolymer is a chain of repetitive units (monomers) that are almost identical, but they differ in their degree of affinity for certain solvents. This difference leads to striking phenomena when the polymer fluctuates in a nonhomogeneous medium, for example, made of two solvents separated by an interface. One may observe, for instance, the localization of the polymer at the interface between the two solvents. A discrete model of such system, based on the simple symmetric random walk on $\mathbb {Z}$, has been investigated in [Bolthausen and den Hollander, Ann. Probab. 25 (1997), 1334-1366], notably in the weak polymer-solvent coupling limit, where the convergence of the discrete model toward a continuum model, based on Brownian motion, has been established. This result is remarkable because it strongly suggests a universal feature of copolymer models. In this work, we prove that this is indeed the case. More precisely, we determine the weak coupling limit for a general class of discrete copolymer models, obtaining as limits a one-parameter [$\alpha \in(0,1)$] family of continuum models, based on $\alpha$-stable regenerative sets.Comment: Published in at http://dx.doi.org/10.1214/10-AOP546 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Section: Introductionmentioning

confidence: 94%

The weak coupling limit of disordered copolymer models

Caravenna¹,

Giacomin²

2010

Ann. Probab.

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“…A related coarse-graining result is proved in Pétrélis [91] for a copolymer model with additional random pinning in a finite layer around the interface (of the type considered in Section 4). It is shown that the effect of the disorder in the layer vanishes in the weak interaction limit, i.e., only the disorder along the copolymer is felt in the weak interaction limit.…”

Section: Weak Interaction Limitmentioning

confidence: 84%

Lectures on random polymers

Caravenna,

Hollander,

Pétrélis

2011

Preprint

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These lecture notes are a guided tour through the fascinating world of polymer chains interacting with themselves and/or with their environment. The focus is on the mathematical description of a number of physical and chemical phenomena, with particular emphasis on phase transitions and space-time scaling. The topics covered, though only a selection, are typical for the area. Sections 1-3 describe models of polymers without disorder, Sections 4-6 models of polymers with disorder. Appendices A-E contain tutorials in which a number of key techniques are explained in more detail. ContentsForeword 319 1. Background, model setting, free energy, two basic models 320 2. Polymer collapse 328 3. A polymer near a homogeneous interface 337 4. A polymer near a random interface 345 5. A copolymer interacting with two immiscible fluids 352 6. A polymer in a random potential 362 Appendix A. Tutorial 1 369 Appendix B. Tutorial 2 371 Appendix C. Tutorial 3 375 Appendix D. Tutorial 4 379 Appendix E. Tutorial 5 385 References 389 ForewordThese notes are based on six lectures by Frank den Hollander and five tutorials by Francesco Caravenna and Nicolas Pétrélis. The final manuscript was prepared jointly by the three authors. A large part of the material is drawn from the monographs by Giambattista Giacomin [55] and Frank den Hollander [70]. Links are

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“…This result was first proved in [7] in the special case of the basic model of section 1, i.e., for the discrete copolymer model based on the simple random walk on Z, corresponding to α = 1 2 (in [17] one can find an argument to relax the assumption in [7] of binary charges and in [25] the case with adsorption is treated, cf. the end of section 1.5).…”

Section: Continuum Model and Weak Coupling Limitmentioning

confidence: 92%

“…Finally, it is natural to wonder what happens when a pinning interaction is added to the copolymer model, that is when not only the solvents are selective, but something special goes on at the interface (thus taking into account for example the lack of sharpness of the interface or the fact that impurities could be trapped at the interface). There are works on this model, often called copolymer with adsorption (see for example [25,27,34]), but the understanding is very limited: we refer to [15, § 6.3.2] for a detailed overview on this issue.…”

Section: The Critical Behavior and A Word About Pinning Modelsmentioning

confidence: 99%

Copolymers at Selective Interfaces: Settled Issues and Open Problems

Caravenna

Giacomin

Toninelli

2011

Probability in Complex Physical Systems

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PrefaceProbabilistic approaches have played a prominent role in the study of complex physical systems for more than 30 years. Two outstanding protagonists of this approach are Jürgen Gärtner and Erwin Bolthausen, to whom this volume is dedicated. Each of them was honored with a workshop in 2010; these took place at Technische Universität Berlin, where they both worked for decades. The conferences were devoted to the most important aspects of their interests: 'Random media' and 'Probabilistic techniques in complex physical systems'. They were organized by the DFG Research Unit FOR718 Analysis and Stochastics in Complex Physical Systems on the occasion of Jürgen's 60th birthday and Erwin's 65th birthday.Jürgen and Erwin have been recognized for decades as outstanding experts in the probabilistic treatment, spiced with a dash of analysis, of problems in statistical mechanics and related fields. Their high esteem and profound impact are reflected by their great number of students and collaborators and by their large number of invitations to conferences, editorships, etc. over the years.Erwin started his career with various distributional limit results of central limit and martingale type, but soon turned to problems coming from large-deviation analysis, like Laplace approximations and the maximum entropy principle. One of the main types of problems that accompanied his career for decades are intricate questions about the extremal behavior of the volume of the path of a random walk or a Wiener sausage and of the intersection of two independent such objects. Here he has derived a number of striking and deep results over the years. Another core area of his research, which is closely related, is the description of paths under the influence of a self-attracting or self-repellent force, partially motivated by the polaron problem. In particular, Erwin derived several fundamental properties of polymers with various kinds of interactions. His results also had a strong influence on the understanding of interface models with gradient-type interactions. Some of his favorite subjects in recent years have been random walks in random environments, and spin glasses and the little-understood phenomenon of ultrametricity.Jürgen was educated within the Russian school in the 1970s, pioneering the application of large deviation analysis to various models in statistical mechanics. v vi Preface

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Copolymer at selective interfaces and pinning potentials: Weak coupling limits

Cited by 5 publications

References 20 publications

The weak coupling limit of disordered copolymer models

The weak coupling limit of disordered copolymer models

Lectures on random polymers

Copolymers at Selective Interfaces: Settled Issues and Open Problems

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