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

Poisat, Julien

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, pp. 456-482.

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.

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.

[1] K. S. Alexander. The effect of disorder on polymer depinning transitions. Comm. Math. Phys. 279 (2008) 117-146. | MR | Zbl

[2] A. E. Allahverdyan, Z. S. Gevorkian, C.-K. Hu and M.-C. Wu. Unzipping of DNA with correlated base sequence. Phys. Rev. E 69 (2004) 061908.

[3] S. Asmussen. Applied Probability and Queues, 2nd edition. Applications of Mathematics (New York) 51. Springer, New York, 2003. | MR | Zbl

[4] F. Caravenna, G. Giacomin and L. Zambotti. A renewal theory approach to periodic copolymers with adsorption. Ann. Appl. Probab. 17 (2007) 1362-1398. | MR | Zbl

[5] X. Y. Chen, L. J. Bao, J. Y. Mo and Y. Wang. Characterizing long-range correlation properties in nucleotide sequences. Chinese Chemical Letters 14 (2003) 503-504.

[6] I. P. Cornfeld, S. V. Fomin and Y. G. Sinaĭ. Ergodic Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 245. Springer, New York, 1982. | MR | Zbl

[7] R. A. Doney. One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Related Fields 107 (1997) 451-465. | MR | Zbl

[8] J. Doob. Stochastic Processes. Wiley Classics Library Edition. Wiley, New York, 1990. | MR | Zbl

[9] A. Garsia and J. Lamperti. A discrete renewal theorem with infinite mean. Comment. Math. Helv. 37 (1962/1963) 221-234. | MR | Zbl

[10] G. Giacomin. Random Polymer Models. Imperial College Press, London, 2007. | MR | Zbl

[11] G. Giacomin. Renewal sequences, disordered potentials, and pinning phenomena. In Spin Glasses: Statics and Dynamics 235-270. Progr. Probab. 62. Birkhäuser, Basel, 2009. | MR | Zbl

[12] J.-H. Jeon, P. J. Park and W. Sung. The effect of sequence correlation on bubble statistics in double-stranded DNA. Journal of Chemical Physics 125 (2006) article 164901.

[13] H. Lacoin. The martingale approach to disorder irrelevance for pinning models. Electron. Commun. Probab. 15 (2010) 418-427. | MR | Zbl

[14] C.-K. Peng, S. V. Buldyrev, A. L. Goldberger, S. Havlin, F. Sciortino, M. Simons and H. E. Stanley. Long-range correlations in nucleotide sequences. Nature 356 (1992) 168-170.

[15] E. Seneta. Non-Negative Matrices and Markov Chains. Springer Series in Statistics. Springer, New York, 2006. | MR | Zbl

[16] F. Spitzer. Principles of Random Walks, 2nd edition. Grad. Texts in Math. 34. Springer, New York, 1976. | MR | Zbl

[17] J. M. Steele. Kingman's subadditive ergodic theorem. Ann. Inst. Henri Poincaré Probab. Stat. 25 (1989) 93-98. | Numdam | MR | Zbl

[18] F. L. Toninelli. A replica-coupling approach to disordered pinning models. Comm. Math. Phys. 280 (2008) 389-401. | MR | Zbl

[19] F. L. Toninelli. Localization transition in disordered pinning models. In Methods of Contemporary Mathematical Statistical Physics 129-176. Lecture Notes in Math. Springer, Berlin, 2009. | MR | Zbl