Continuous model for homopolymers

Cranston, M.; Koralov, Leonid; Molchanov, Stanislav; Vainberg, Boris

doi:10.1016/j.jfa.2008.07.019

Cited by 61 publications

(75 citation statements)

References 6 publications

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“…We point out that this kind of questions have been answered in depth in the case of the standard wetting model, that is formally in the a = 0 case, and that the problem of extending these results to the case where the defects are in a strip has been posed in [16], Chapter 2. Note that a closely related pinning model in continuous time has been considered and resolved in [10]; we stress however that their techniques are very peculiar to the continuous time setup. is an interface separating a dry zone from a wet zone.…”

Section: Definition Of the Modelmentioning

confidence: 99%

The scaling limits of a heavy tailed Markov renewal process

Sohier¹

2013

Ann. Inst. H. Poincaré Probab. Statist.

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In this paper we consider heavy tailed Markov renewal processes and we prove that, suitably renormalised, they converge in law towards the α-stable regenerative set. We then apply these results to the strip wetting model which is a random walk S constrained above a wall and rewarded or penalized when it hits the strip [0, ∞) × [0, a] where a is a given positive number. The convergence result that we establish allows to characterize the scaling limit of this process at criticality. Résumé.Dans cet article, nous considérons des processus de renouvellement markovien à queues lourdes. Nous montrons que, convenablement renormalisés, ils convergent vers l'ensemble régénératif d'indice α. Nous appliquons ces résultats à un modèle d'accrochage dans une bande. Dans ce modèle, une marche aléatoire S, contrainte à rester au-dessus d'un mur, est récompensée ou pénalisée lorsqu'est atteinte la bande [0, ∞) × [0, a] où a est un réel strictement positif. La convergence que nous établissons permet de caractériser les limites d'échelle de ce modèle au point critique. MSC: 60F77; 60K15; 60K20; 60K05; 82B27 Keywords: Heavy tailed Markov renewals processes; Scaling limits; Fluctuation theory for random walks; Regenerative sets k(x, y) := 1 μ(dy) n≥1 K x,dy (n) = n≥1 k x,y (n)is well defined. 484J. Sohier 2. Then sample J as a Markov chain on E starting from some initial point J 0 = x 0 ∈ E and with transition kernel3. Finally sample the increments (τ i − τ i−1 ) i≥1 as a sequence of independent, but not identically distributed random variables according to the conditional law:Markov renewal processes have been introduced independently by Lévy [18] and Smith [21], and their basic properties have been studied by Pyke [19], Cinlar [9] and Pyke and Schauffele [20] among others. A modern reference is Asmussen [1], VII, 4, which describes some applications related to these processes, in particular in the field of queuing theory. More recently, they have been widely used as a convenient tool to describe phenomena arising in models issued from statistical mechanics, and more particularly in models related to pinning models, such as the periodic wetting model ([8] and the monograph [16], Chapter 3) or the 1 + 1 dimensional field with Laplacian interaction ([4] and [5]).We will show our results in the case where the kernel K satisfies the following assumptions.Assumption 1.1. We make the following assumptions on the kernel K:

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Section: Definition Of the Modelmentioning

confidence: 99%

The scaling limits of a heavy tailed Markov renewal process

Sohier¹

2013

Ann. Inst. H. Poincaré Probab. Statist.

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“…It is known , in the theory of BRWs the problem on the spectrum of a evolutionary operator of mean particle numbers plays an important role. Resolvent analysis of such operators (Cranston M. et al, 2009) has allowed to investigate BRWs with large deviation . The limit theorems on asymptotic behavior of the Green's function for transition probabilities were established.…”

Section: Ergodic Propertymentioning

confidence: 99%

Simulating Correlated Ordinal and Discrete Variables with Assigned Marginal Distributions

Barbiero

Ferrari

2014

Springer Proceedings in Mathematics &Amp; Statistics

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“…Simon , Cranston et al , etc., considered the same problem for Brownian motion. They derived an exact asymptotic behavior of

e_{μ} (t, x)

in case that μ is a nonnegative, continuous function with compact support.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, they showed that the asymptotic behavior depends on the dimension d and is at most proportional to t . For

α = 2

, Theorem is equivalent to the result of or .…”

Section: Introductionmentioning

confidence: 99%

Large time asymptotics of Feynman–Kac functionals for symmetric stable processes

Takeda

Wada

2016

Mathematische Nachrichten

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Let μ be a positive Kato measure on Rd associated with the Green kernel of the transient symmetric α‐stable process, the Markov process with generator (−Δ)α/2 (d>α). Let Atμ be the positive continuous additive functional in the Revuz correspondence with μ. If, in addition, μ is of compact support, we give exact large time asymptotics of the expectation of the Feynman–Kac functional, exp(Atμ).

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Continuous model for homopolymers

Cited by 61 publications

References 6 publications

The scaling limits of a heavy tailed Markov renewal process

The scaling limits of a heavy tailed Markov renewal process

Simulating Correlated Ordinal and Discrete Variables with Assigned Marginal Distributions

Large time asymptotics of Feynman–Kac functionals for symmetric stable processes

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