Localization Transition in Disordered Pinning Models

Toninelli, Fabio Lucio

doi:10.1007/978-3-540-92796-9_3

Cited by 7 publications

(8 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There has been substantial investigation of pinning models in which disorder is present only in the defect line {0} × N; see ( [17,18] and [32]) for surveys. In such models (which we call pinning models with defect-line potential), the energy gains from pinning compete only with the entropy loss inherent in the class of pinned paths.…”

Section: Introductionmentioning

confidence: 99%

Directed polymers in a random environment with a defect line

Alexander

Yildirim

2015

Electron. J. Probab.

View full text Add to dashboard Cite

We study the depinning transition of the $1+1$ dimensional directed polymer in a random environment with a defect line. The random environment consists of i.i.d. potential values assigned to each site of $\mathbb{Z}^2$; sites on the positive axis have the potential enhanced by a deterministic value $u$. We show that for small inverse temperature $\beta$ the quenched and annealed free energies differ significantly at most in a small neighborhood (of size of order $\beta$) of the annealed critical point $u_c^a=0$. For the case $u=0$, we show that the difference between quenched and annealed free energies is of order $\beta^4$ as $\beta\to 0$, assuming only finiteness of exponential moments of the potential values, improving existing results which required stronger assumptions.Comment: 22 pages. Changes to Proposition 3.8 make Proposition 4.1 unnecessary; minor corrections mad

show abstract

Section: Introductionmentioning

confidence: 99%

Directed polymers in a random environment with a defect line

Alexander

Yildirim

2015

Electron. J. Probab.

View full text Add to dashboard Cite

show abstract

“…When disorder is i.i.d, (5) becomes h c (β) ≥ h c (0) − β 2 /2 := h a c (β) and the question of knowing whether this is an equality was studied in several papers and monographs (for example, [9], [19], [10] and references therein) where we learn that the answer depends on the values of α and β.…”

Section: Annealed Free Energy and Annealed Boundmentioning

confidence: 99%

“…but one has to justify the interchange of the integration and the asymptotic equivalent. Indeed, from the upper bound (19) in Lemma 5.1, one can apply the Dominated Convergence Theorem, because E(θ c ) < ∞ (it is not hard to see that the tail of θ decays exponentially fast). We prove the last point of the proposition.…”

Section: Intersection Of Markov Renewal Processesmentioning

confidence: 99%

See 1 more Smart Citation

On quenched and annealed critical curves of random pinning model with finite range correlations

Poisat¹

2013

Ann. Inst. H. Poincaré Probab. Statist.

View full text Add to dashboard Cite

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 annealed model are particular Markov renewal processes. We consider the intersection of two replicas of this process to prove a result of disorder irrelevance (i.e. quenched and annealed critical curves as well as exponents coincide) via the method of second moment

show abstract

“…This paper deals with two pinning models with Markov disorder, and is organized as follows: in a first part we will recall notations and general facts on the model, and give an overview of results dealing with the critical curve of the phase diagram (for a more detailed account on pinning models, the reader can refer to the monographs [10], [8,Ch. 7,11] and the survey papers [11], [18]). We then introduce the two models, which will be developed in one section each.…”

Section: Introductionmentioning

confidence: 99%

Two Pinning Models with Markov disorder

Poisat

2010

Preprint

View full text Add to dashboard Cite

Disordered pinning models deal with the (de)localization transition 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 speaking, disorder is more and more correlated. In this case we are able to give the limit of the averaged quenched free energy, therefore providing the full phase diagram picture, and the number of critical points is related to the number of states of the Markov chain. We also mention that the question of pinning in correlated disorder appears in the context of DNA denaturation.

show abstract

Localization Transition in Disordered Pinning Models

Cited by 7 publications

References 34 publications

Directed polymers in a random environment with a defect line

Directed polymers in a random environment with a defect line

On quenched and annealed critical curves of random pinning model with finite range correlations

Two Pinning Models with Markov disorder

Contact Info

Product

Resources

About