In this article, I study the localization transition of an hydrophobic homopolymer in interaction with an interface between oil and water. To that aim I consider a model in which the trajectories of a simple random walk play the role of the possible configurations of the polymer. The chain gains an energetic factor for every monomer it puts in the oil and receives a random price of positive average each time it touches the origin. So a competition arises between this two effects to know wether the chain is delocalized in the oil or if it stays localized in the neighborhood of the interface. As usual a critical curve divides the phase spaces in a localized and a delocalized area, and the point of my paper is to develop a new strategy to bound this critical curve by below. To achieve this result I take into account the fact that the polymer can target the sites where it comes back to the origin to receive prices as high as possible.
In this article we discuss a set of geometric ideas which shed some light on the question of directed polymer pinning in the presence of bulk disorder. Differing from standard methods and techniques, we transform the problem to a particular dependent percolative system and relate the pinning transition to a percolation transition.
Large deviations principle is obtained for terminating multidimensional compound renewal processes. We also obtained the asymptotic of large deviations for the case when a Gibbs change of the original probability measure takes place. The random processes mentioned in the paper are widely used in polymer pinning models.
This paper studies a polymer chain in the vicinity of a linear interface separating two immiscible solvents. The polymer consists of random monomer types, while the interface carries random charges. Both the monomer types and the charges are given by i.i.d. sequences of random variables. The configurations of the polymer are directed paths that can make i.i.d. excursions of finite length above and below the interface. The Hamiltonian has two parts: a monomer-solvent interaction ("copolymer") and a monomer-interface interaction ("pinning"). The quenched and the annealed version of the model eac…
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.
In this paper we look at the pinning of a directed polymer by a one-dimensional linear interface carrying random charges. There are two phases, localized and delocalized, depending on the inverse temperature and on the disorder bias. Using quenched and annealed large deviation principles for the empirical process of words drawn from a random letter sequence according to a random renewal process [Birkner, Greven and den Hollander, Probab. Theory Related Fields 148 (2010) 403-456], we derive variational formulas for the quenched, respectively, annealed critical curve separating the two phases. T…
We consider a model for a polymer chain interacting with a sequence of equispaced flat interfaces through a pinning potential. The intensity $δ\in \mathbb {R}$ of the pinning interaction is constant, while the interface spacing $T=T_N$ is allowed to vary with the size $N$ of the polymer. Our main result is the explicit determination of the scaling behavior of the model in the large $N$ limit, as a function of $(T_N)_N$ and for fixed $δ>0$. In particular, we show that a transition occurs at $T_N=O(\log N)$. Our approach is based on renewal theory.
We consider the Glauber dynamics for model of polymer interacting with a substrate or wall. The state space is the set of one-dimensional nearest-neighbor paths on $\mathbb{Z}$ with nonnegative integer coordinates, starting at $0$ and coming back to $0$ after $L$ ($L\in 2\mathbb{N}$) steps and the Gibbs weight of a path $ξ=(ξ_x)^{L}_{x=0}$ is given by $λ^{\mathcal{N}(ξ)}$, where $λ\geq 0$ is a parameter which models the intensity of the interaction with the substrate and $\mathcal{N}(ξ)$ is the number of zeros in $ξ$. The dynamics we consider proceeds by updating $ξ_x$ with rate one for each $…
We consider a directed polymer interacting with a diluted pinning potential restricted to a line. We characterize explicitely 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.
We consider a directed polymer of length $N$ interacting with a linear interface. The monomers carry i.i.d. random charges $(ω_i)_{i=1}^N$ taking values in $\mathbb{R}$ with mean zero and variance one. Each monomer $i$ contributes an energy $(βω_i-h)φ(S_i)$ to the interaction Hamiltonian, where $S_i \in \mathbb{Z}$ is the height of monomer $i$ with respect to the interface, $φ: \mathbb{Z} \to [0,\infty)$ is the interaction potential, $β\in [0,\infty)$ is the inverse temperature, and $h \in \mathbb{R}$ is the charge bias parameter. The configurations of the polymer are weighted according to the…
We consider the stochastic evolution of a 1+1-dimensional interface (or polymer) in presence of a substrate. This stochastic process is a dynamical version of the homogeneous pinning model. We start from a configuration far from equilibrium: a polymer with a non-trivial macroscopic height profile, and look at the evolution of a space-time rescaled interface. In two cases, we prove that this rescaled interface has a scaling limit on the diffusive scale (space rescaled by $L$ in both dimensions and time rescaled by $L^2$ where $L$ denotes the length of the interface) which we describe: when the …
We consider the model of a directed polymer pinned to a line of i.i.d. random charges, and focus on the interior of the delocalized phase. We first show that in this region, the partition function remains bounded. We then prove that for almost every environment of charges, the probability that the number of contact points in [0,n] exceeds c log(n) tends to 0 as n tends to infinity. Our proofs rely on recent results of Birkner, Greven, den Hollander (2010) and Cheliotis, den Hollander (2010).
We consider a statistical mechanics model for biopolymers. Sophisticated polymer chains, such as DNA, have stiffness when they stretch chains. The Laplacian interaction is used to describe the stiffness. Also, the surface between two media has an attraction force, and the force will pull the chain back to the surface. In this paper, we deal with the random potentials when the monomers interact with the random media. Although these models are different from the pinning models studied before, the result about the gap between the annealed critical point and the quenched critical point stays the s…
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. Disorder is introduced by, for example, 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 of its monomers at 0 with high probability. This critical point may differ for the quenched, annealed and deterministic cases. We show that self-averaging occurs, meani…
In this paper, we study a model of directed polymers in random environment, where the environment is restricted to a time-space tube whose spatial width grows polynomially with time. It can be viewed as an interpolation between the disordered pinning model and the classic directed polymer model. We prove weak coupling limits for the directed polymer partition functions in such tube environments in all dimensions, as the inverse temperature vanishes at a suitable rate. As the tube width varies, transitions between regimes of disorder irrelevance, marginal relevance and disorder relevance are ob…
These notes are devoted to the statistical mechanics of directed polymers interacting with one-dimensional 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 denaturati…
We consider a simple random walk of length $N$, denoted by $(S_{i})_{i\in \{1,...,N\}}$, and we define $(w_i)_{i\geq 1}$ a sequence of centered i.i.d. random variables. For $K\in\N$ we define $((γ_i^{-K},...,γ_i^K))_{i\geq 1}$ an i.i.d sequence of random vectors. We set $β\in \mathbb{R}$, $λ\geq 0$ and $h\geq 0$, and transform the measure on the set of random walk trajectories with the Hamiltonian $λ\sum_{i=1}^{N} (w_i+h) \sign(S_i)+β\sum_{j=-K}^{K}\sum_{i=1}^{N} γ_{i}^{j} \boldsymbol{1}_{\{S_{i}=j\}}$. This transformed path measure describes an hydrophobic(philic) copolymer interacting with a…
We study a random walk pinning model, where conditioned on a simple random walk Y on Z^d acting as a random medium, the path measure of a second independent simple random walk X up to time t is Gibbs transformed with Hamiltonian -L_t(X,Y), where L_t(X,Y) is the collision local time between X and Y up to time t. This model arises naturally in various contexts, including the study of the parabolic Anderson model with moving catalysts, the parabolic Anderson model with Brownian noise, and the directed polymer model. It falls in the same framework as the pinning and copolymer models, and exhibits …
The membrane model is a Gaussian interface model with a Hamiltonian involving second derivatives of the interface height. We consider the model in dimension $\mathsf{d}\ge4$ under the influence of $δ$-pinning of strength $\varepsilon$. It is known that this pinning potential manages to localize the interface for any $\varepsilon>0$. We refine this result by establishing the $\varepsilon$-dependence of the variance and of the exponential decay rate of the covariances for small $\varepsilon$ (similar to the corresponding results for the discrete Gaussian free field by Bolthausen-Velenik). We als…
In this paper we consider a model which describes a polymer chain interacting with an infinity of equi-spaced linear interfaces. The distance between two consecutive interfaces is denoted by T = T_N 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 a previous paper, showing that a transition occurs when T_N \approx 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 …
We consider a polymer with configuration modeled by the path of a Markov chain, interacting with a potential $u+V_n$ which the chain encounters when it visits a special state 0 at time $n$. The disorder $(V_n)$ is a fixed realization of an i.i.d. sequence. The polymer is pinned, i.e. the chain spends a positive fraction of its time at state 0, when $u$ exceeds a critical value. We assume that for the Markov chain in the absence of the potential, the probability of an excursion from 0 of length $n$ has the form $n^{-c}ϕ(n)$ with $c \geq 1$ and $ϕ$ slowly varying. Comparing to the corresponding …
Disordered pinning models deal with the (de)localization tran- sition of a polymer in interaction with a heterogeneous interface. In this paper, we focus on two models where the inhomogeneities at the interface are not independent but given by an irreducible Markov chain on a finite state space. In the first model, using Markov renewal tools, we give an expression for the annealed critical curve in terms of a Perron-Frobenius eigenvalue, and provide examples where exact computations are possible. In the second model, the transition matrix vary with the size of the system so that, roughly speak…
For a model of a driven interface in an elastic medium with random obstacles we prove existence of a stationary positive supersolution at non-vanishing driving force. This shows the emergence of a rate independent hysteresis through the interaction of the interface with the obstacles, despite a linear (force=velocity) microscopic kinetic relation. We also prove a percolation result, namely the possibility to embed the graph of an only logarithmically growing function in a next-nearest neighbor site-percolation cluster at a non-trivial percolation threshold.
We consider statistical mechanics models of continuous height effective interfaces in the presence of a delta-pinning at height zero. There is a detailed mathematical understanding of the depinning transition in 2 dimensions without disorder. Then the variance of the interface height w.r.t. the Gibbs measure stays bounded uniformly in the volume for any positive pinning force and diverges like the logarithm of the pinning force when it tends to zero. How does the presence of a quenched disorder term in the Hamiltonian modify this transition? We show that an arbitarily weak random field term is…
This paper focuses on directed polymers pinned at a disordered and correlated interface. We assume that the disorder sequence is a q-order moving average and show that the critical curve of the annealed model can be expressed in terms of the Perron-Frobenius eigenvalue of an explicit transfer matrix, which generalizes the annealed bound of the critical curve for i.i.d. disorder. We provide explicit values of the annealed critical curve for $q=1$, 2 and a weak disorder asymptotic in the general case. Following the renewal theory approach of pinning, the processes arising in the study of the ann…
We study the so-called pinning model, which describes the behavior of a Markov chain interacting with a distinguished state. The interaction depends on an external source of randomness, called disorder, which can attract or repel the Markov chain path. We focus on the case when the disorder is heavy-tailed, with infinite mean, while the return times of the Markov chain have a stretched-exponential distribution. We prove that the set of times at which the Markov chain visits the distinguished state, suitably rescaled, converges in distribution to a limit set, which depends only on the disorder …
We consider two models for biopolymers, the $\nabla$ interaction and the $Δ$ one, both with the Gaussian potential in the random environment. A random field $φ:{0,1,...,N}\rightarrow \Bbb{R}^d$ represents the position of the polymer path. The law of the field is given by $\exp(-\sum_i\frac{|\nablaφ_i|^2}{2})$ where $\nabla$ is the discrete gradient, and by $\exp(-\sum_i\frac{|Δφ_i|^2}{2})$ where $Δ$ is the discrete Laplacian. For every Gaussian potential $\frac{|\cdot|^2}{2}$, a random charge is added as a factor: $(1+βω_i)\frac{|\cdot|^2}{2}$ with $\Bbb{P}(ω_i=\pm 1)=1/2$ or $\exp(βω_i)\frac{…
We introduce the pinning model on a quenched renewal, which is an instance of a (strongly correlated) disordered pinning model. The potential takes value 1 at the renewal times of a quenched realization of a renewal process $σ$, and $0$ elsewhere, so nonzero potential values become sparse if the gaps in $σ$ have infinite mean. The "polymer" -- of length $σ_N$ -- is given by another renewal $τ$, whose law is modified by the Boltzmann weight $\exp(β\sum_{n=1}^N \mathbf{1}_{\{σ_n\inτ\}})$. Our assumption is that $τ$ and $σ$ have gap distributions with power-law-decay exponents $1+α$ and $1+\tilde…
We prove that for a class of massless $\nablaϕ$ interface models on $\Ztwo$ an introduction of an arbitrary small pinning self-potential leads to exponential decay of correlation, or, in other words, to creation of mass.
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.
We consider paths of a one-dimensional simple random walk conditioned to come back to the origin after L steps (L an even integer). In the 'pinning model' each path ηhas a weight λ^{N(η)}, where λ>0 and N(η) is the number of zeros in η. When the paths are constrained to be non-negative, the polymer is said to satisfy a hard-wall constraint. Such models are well known to undergo a localization/delocalization transition as the pinning strength λis varied. In this paper we study a natural 'spin flip' dynamics for these models and derive several estimates on its spectral gap and mixing time. In pa…
We prove an identity in distribution between two kinds of partition functions for the log-gamma directed polymer model: (1) the point-to-point partition function in a quadrant, (2) the point-to-line partition function in an octant. As an application, we prove that the point-to-line free energy of the log-gamma polymer in an octant obeys a phase transition depending on the strength of the noise along the boundary. This transition of (de)pinning by randomness was first predicted in physics by Kardar in 1985 and proved rigorously for zero temperature models by Baik and Rains in 2001. While it is …
We investigate the localization transition for a simple model of interface which interacts with an inhomonegeous defect plane. The interface is modeled by the graph of a function $ϕ: \mathbb Z^2 \to \mathbb Z$,and the disorder is given by a fixed realization of a field of IID centered random variables$(ω_x)_{x\in \mathbb Z^2}$. The Hamiltonian of the system depends on three parameters $α,β>0$ and $h\in \mathbb R$ which determine respectively the intensity of nearest neighbor interaction the amplitude of disorder and the mean value of the interaction with the substrate, and is given by the expr…
