Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Julien Poisat

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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 $q=1$ and $q=2$ 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.


Dans cet article nous étudions le modèle des polymères dirigés accrochés á une interface désordonnée et corrélée. Nous supposons que le désordre est une moyenne mobile d’ordre $q$ et nous montrons que la courbe critique du modèle annealed peut s’exprimer en fonction de la valeur propre de Perron–Frobenius d’une matrice de transfert explicite, ce qui généralise la borne annealed de la courbe critique dans le cas d’un désordre i.i.d. Nous donnons des valeurs explicites de la courbe annealed pour $q=1$ et $q=2$ et un équivalent á faible désordre dans le cas général. Du point de vue de la théorie du renouvellement, les processus qui interviennent dans l’étude du modèle annealed sont des processus de renouvellement markoviens particuliers. Nous considérons l’intersection de deux répliques de ces processus pour prouver un résultat de non-pertinence du désordre (les courbes ainsi que les exposants critiques annealed et quenched coïncident) via la méthode du moment d’ordre deux.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 2 (2013), 456-482.

First available in Project Euclid: 16 April 2013

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Zentralblatt MATH identifier

Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 60K37: Processes in random environments 60K05: Renewal theory

Polymer models Pinning Annealed model Disorder irrelevance Correlated disorder Renewal process Markov renewal process Intersection of renewal processes Perron–Frobenius theory Subadditivity


Poisat, Julien. On quenched and annealed critical curves of random pinning model with finite range correlations. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 2, 456--482. doi:10.1214/11-AIHP446.

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