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32
IVY ON THE CEILING: FIRSTORDER POLYMER DEPINNING TRANSITIONS WITH QUENCHED DISORDER
, 2006
"... Abstract. We consider a polymer, with monomer locations modeled by the trajectory of an underlying Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. Disorder is introduced by having the interaction vary from one monomer to another, as a ..."
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Abstract. We consider a polymer, with monomer locations modeled by the trajectory of an underlying Markov chain, in the presence of a potential that interacts with the polymer when it visits a particular site 0. Disorder is introduced by having the interaction vary from one monomer to another, as a constant u plus i.i.d. mean0 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. When the excursions of the underlying chain have a finite mean but no finite exponential moment, it is known [2] that the depinning transition (more precisely, the contact fraction) in the corresponding annealed system is discontinuous. One generally expects the presence of disorder to smooth transitions, and it is known [6] that when the excursion length distribution has powerlaw tails, the quenched system has a continuous transition even if the annealed system does not. We show here that when the underlying chain is transient but the finite part of the excursion length distribution has exponential tails, then the depinning transition is discontinuous even in the quenched system, and the quenched and annealed critical points are strictly different. By contrast, in the recurrent case, the depinning behavior depends on the subexponential prefactors on the exponential decay of the excursion length distribution, and when these prefactors decay with an appropriate power law, the quenched transition is continuous even though the annealed one is not. 1.
Correlation lengths for random polymer models and for some renewal sequences, preprint
, 2006
"... Abstract. We consider models of directed polymers interacting with a onedimensional defect line on which random charges are placed. More abstractly, one starts from renewal sequence on Z and gives a random (sitedependent) reward or penalty to the occurrence of a renewal at any given point of Z. Th ..."
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Abstract. We consider models of directed polymers interacting with a onedimensional defect line on which random charges are placed. More abstractly, one starts from renewal sequence on Z and gives a random (sitedependent) reward or penalty to the occurrence of a renewal at any given point of Z. These models are known to undergo a delocalizationlocalization transition, and the free energy f vanishes when the critical point is approached from the localized region. We prove that the quenched correlation length ξ, defined as the inverse of the rate of exponential decay of the twopoint function, does not diverge faster than 1/f. We prove a lower bound also for the rate of exponential decay of the disorderaveraged twopoint function. We discuss how, in the particular case where disorder is absent, this result can be seen as a refinement of the classical renewal theorem, for a specific class of renewal sequences.
LOCALIZATION TRANSITION IN DISORDERED PINNING MODELS. EFFECT OF RANDOMNESS ON THE CRITICAL PROPERTIES.
, 2007
"... These notes are devoted to the statistical mechanics of directed polymers interacting with onedimensional spatial defects. We are interested in particular in the situation where frozen disorder is present. These polymer models undergo a localization/delocalization transition. There is a large (bio ..."
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These notes are devoted to the statistical mechanics of directed polymers interacting with onedimensional spatial defects. We are interested in particular in the situation where frozen disorder is present. These polymer models undergo a localization/delocalization transition. There is a large (bio)physics literature on the subject since these systems describe, for instance, the statistics of thermally created loops in DNA double strands and the interaction between (1 + 1)dimensional interfaces and disordered walls. In these cases the transition corresponds, respectively, to the DNA denaturation transition and to the wetting transition. More abstractly, one may see these models as random and inhomogeneous perturbations of renewal processes. The last few years have witnessed a great progress in the mathematical understanding of the equilibrium properties of these systems. In particular, many rigorous results about the location of the critical point, about critical exponents and path properties of the polymer in the two thermodynamic phases (localized and delocalized) are now available. Here, we will focus on some aspects of this topic in particular, on the nonperturbative effects of disorder. The mathematical tools employed range from renewal theory to large deviations and, interestingly, show tight connections with techniques developed recently in the mathematical study of mean field spin glasses.
Pinning by a sparse potential
, 2005
"... We consider a directed polymer interacting with a diluted pinning potential restricted to a line. We characterize explicitly the set of disorder configurations that give rise to localization of the polymer. We study both relevant cases of dimension 1+1 and 1+2. We also discuss the case of massless e ..."
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We consider a directed polymer interacting with a diluted pinning potential restricted to a line. We characterize explicitly the set of disorder configurations that give rise to localization of the polymer. We study both relevant cases of dimension 1+1 and 1+2. We also discuss the case of massless effective interface models in dimension 2+1.
The quenched critical point of a diluted disordered polymer model
, 2007
"... Abstract. We consider a model for a polymer interacting with an attractive wall through a random sequence of charges. We focus on the socalled diluted limit, when the charges are very rare but have strong intensity. In this regime, we determine the quenched critical point of the model, showing that ..."
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Abstract. We consider a model for a polymer interacting with an attractive wall through a random sequence of charges. We focus on the socalled diluted limit, when the charges are very rare but have strong intensity. In this regime, we determine the quenched critical point of the model, showing that it is different from the annealed one. The proof is based on a rigorous renormalization procedure. Applications of our results to the problem of a copolymer near a selective interface are discussed. 1.
Section de Math
"... Cette thèse est consacrée à l’étude de plusieurs modèles de Mécanique Statistique qui peuvent être vus comme des modèles d’interfaces. Dans la première partie, nous étudions l’ensemble des mesures de Gibbs en volume infini correspondant aux modèles d’Ising et de Potts dans le régime de coexistence d ..."
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Cette thèse est consacrée à l’étude de plusieurs modèles de Mécanique Statistique qui peuvent être vus comme des modèles d’interfaces. Dans la première partie, nous étudions l’ensemble des mesures de Gibbs en volume infini correspondant aux modèles d’Ising et de Potts dans le régime de coexistence de phases, i.e. sous la température critique. Le premier résultat est un raffinement du célèbre théorème indépendamment établi par Aizenman et Higuchi durant la fin des années 70, affirmant que toutes les mesures de Gibbs en volume infini correspondant au modèle d’Ising plus proche voisins sur Z^2 sont combinaisons convexes de P+ et P, les deux phases pures du modèle. Nous présentons une nouvelle approche à ce résultat, et étendons l’approche développée pour le modèle d’Ising, pour prouver que tous les états de Gibbs du modèle de Potts plus proche voisins à q états sur Z^2 sous la température critique sont les combinaisons convexes des q phases monochromatiques. En particulier, ils sont tous invariants sous les translations. Ce résultat était conjecturé par les physiciens [...] COQUILLE, Loren. Flowers, Forests and Fields in physics. Thèse de doctorat: Univ.
Institut Henri Poincaré Probabilités et Statistiques
, 2009
"... Abstract. We consider a simple random walk of length N, denoted by (S i ) i∈{1, ..."
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Abstract. We consider a simple random walk of length N, denoted by (S i ) i∈{1,
of random interfaces∗
"... The study of effective interface models has been quite active recently, with a particular emphasis on the effect of various external potentials (wall, pinning potential,...) leading to localization/delocalization transitions. I review some of the results that have been obtained. In particular, I dis ..."
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The study of effective interface models has been quite active recently, with a particular emphasis on the effect of various external potentials (wall, pinning potential,...) leading to localization/delocalization transitions. I review some of the results that have been obtained. In particular, I discuss pinning by a local potential, entropic repulsion and the (pre)wetting transition, both for models with continuous and discrete heights. This text is based on lecture notes for a minicourse given during the workshop "Topics in Random Interfaces and Directed
DOI: 10.1214/154957804100000000 Localization and delocalization of random interfaces∗
"... url: www.univrouen.fr/LMRS/Persopage/Velenik/ Abstract: The probabilistic study of effective interface models has been quite active in recent years, with a particular emphasis on the effect of various external potentials (wall, pinning potential,...) leading to localization/delocalization transiti ..."
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url: www.univrouen.fr/LMRS/Persopage/Velenik/ Abstract: The probabilistic study of effective interface models has been quite active in recent years, with a particular emphasis on the effect of various external potentials (wall, pinning potential,...) leading to localization/delocalization transitions. I review some of the results that have been obtained. In particular, I discuss pinning by a local potential, entropic repulsion and the (pre)wetting transition, both for models with continuous and discrete heights.