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Pinning and wetting transition for (1+1)dimensional fields with Laplacian interaction
 Ann. Probab
"... Abstract. We consider a random field ϕ: {1,..., N} → R as a model for a linear chain attracted to the defect line ϕ = 0, i.e. the x–axis. The free law of the field is specified by the density exp ` − P i V (∆ϕi) ´ with respect to the Lebesgue measure on R N, where ∆ is the discrete Laplacian and w ..."
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Cited by 25 (4 self)
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Abstract. We consider a random field ϕ: {1,..., N} → R as a model for a linear chain attracted to the defect line ϕ = 0, i.e. the x–axis. The free law of the field is specified by the density exp ` − P i V (∆ϕi) ´ with respect to the Lebesgue measure on R N, where ∆ is the discrete Laplacian and we allow for a very large class of potentials V (·). The interaction with the defect line is introduced by giving the field a reward ε ≥ 0 each time it touches the x–axis. We call this model the pinning model. We consider a second model, the wetting model, in which, in addition to the pinning reward, the field is also constrained to stay nonnegative. We show that both models undergo a phase transition as the intensity ε of the pinning reward varies: both in the pinning (a = p) and in the wetting (a = w) case, there exists a critical value ε a c such that when ε> ε a c the field touches the defect line a positive fraction of times (localization), while this does not happen for ε < ε a c (delocalization). The two critical values are nontrivial and distinct: 0 < ε p c < ε w c < ∞, and they are the only nonanalyticity points of the respective free energies. For the pinning model the transition is of second order, hence the field at ε = ε p c is delocalized. On the other hand, the transition in the wetting model is of first order and for ε = ε w c the field is localized. The core of our approach is a Markov renewal theory description of the field. 1. Introduction and
On the irrelevant disorder regime of pinning models
, 2007
"... Recent results have lead to substantial progress in understanding the role of disorder in the (de)localization transition of polymer pinning models. Notably, there is an understanding of the crucial issue of disorder relevance and irrelevance that, albeit still partial, is now rigorous. In this wor ..."
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Cited by 14 (4 self)
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Recent results have lead to substantial progress in understanding the role of disorder in the (de)localization transition of polymer pinning models. Notably, there is an understanding of the crucial issue of disorder relevance and irrelevance that, albeit still partial, is now rigorous. In this work we exploit interpolation and replica coupling methods to get sharper results on the irrelevant disorder regime of pinning models. In particular, we compute in this regime the first order term in the expansion of the free energy close to criticality, which coincides with the first order of the formal expansion obtained by field theory methods. We also show that the quenched and the quenched averaged correlation length exponents coincide, while in general they are expected to be different. Interpolation and replica coupling methods in this class of models naturally lead to studying the behavior of the intersection of certain renewal sequences and one of the main tools in this work is precisely renewal theory and the study of these intersection renewals.
A functional limit theorem for random walk conditioned to stay nonnegative
 J. Lond. Math. Soc
, 2006
"... There is an increasing use in modelling of processes derived from random walks which are constrained to stay on a halfline; see [10] for a recent example of this. However the phrase "random walk conditioned to stay nonnegative " has at least two di¤erent interpretations. In the …rst we c ..."
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Cited by 13 (2 self)
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There is an increasing use in modelling of processes derived from random walks which are constrained to stay on a halfline; see [10] for a recent example of this. However the phrase "random walk conditioned to stay nonnegative " has at least two di¤erent interpretations. In the …rst we consider the …rst n values of a
H.: Sharp critical behavior for pinning models in a random correlated environment. Stochastic Process
 Appl
, 2012
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Infinite volume limits of polymer chains with periodic charges. Markov Process. Related Fields,
, 2007
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DEPINNING OF A POLYMER IN A MULTIINTERFACE MEDIUM
, 901
"... Abstract. In this paper we consider a model which describes a polymer chain interacting with an infinity of equispaced linear interfaces. The distance between two consecutive interfaces is denoted by T = TN and is allowed to grow with the size N of the polymer. When the polymer receives a positive ..."
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Cited by 4 (2 self)
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Abstract. In this paper we consider a model which describes a polymer chain interacting with an infinity of equispaced linear interfaces. The distance between two consecutive interfaces is denoted by T = TN and is allowed to grow with the size N of the polymer. When the polymer receives a positive reward for touching the interfaces, its asymptotic behavior has been derived in [3], showing that a transition occurs when TN ≈ log N. In the present paper, we deal with the so–called depinning case, i.e., the polymer is repelled rather than attracted by the interfaces. Using techniques from renewal theory, we determine the scaling behavior of the model for large N as a function of {TN}N, showing that two transitions occur, when TN ≈ N 1/3 and when TN ≈ √ N respectively. 1. Introduction and
TIGHTNESS CONDITIONS FOR POLYMER MEASURES
, 2007
"... Abstract. We give sufficient conditions for tightness in the space C([0, 1]) for sequences of probability measures which enjoy a suitable decoupling between zero level set and excursions. Applications of our results are given in the context of (homogeneous, periodic and disordered) random walk model ..."
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Abstract. We give sufficient conditions for tightness in the space C([0, 1]) for sequences of probability measures which enjoy a suitable decoupling between zero level set and excursions. Applications of our results are given in the context of (homogeneous, periodic and disordered) random walk models for polymers and interfaces.