On quenched and annealed critical curves of random pinning model with finite range correlations
Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 2, p. 456-482

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.

DOI : https://doi.org/10.1214/11-AIHP446
Classification:  82B44,  60K37,  60K05
Keywords: polymer models, pinning, annealed model, disorder irrelevance, correlated disorder, renewal process, Markov renewal process, intersection of renewal processes, Perron-Frobenius theory, subadditivity
     author = {Poisat, Julien},
     title = {On quenched and annealed critical curves of random pinning model with finite range correlations},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {2},
     year = {2013},
     pages = {456-482},
     doi = {10.1214/11-AIHP446},
     zbl = {1276.82024},
     mrnumber = {3088377},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2013__49_2_456_0}
Poisat, Julien. On quenched and annealed critical curves of random pinning model with finite range correlations. Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 2, pp. 456-482. doi : 10.1214/11-AIHP446. http://www.numdam.org/item/AIHPB_2013__49_2_456_0/

[1] K. S. Alexander. The effect of disorder on polymer depinning transitions. Comm. Math. Phys. 279 (2008) 117-146. | MR 2377630 | Zbl 1175.82034

[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 1978607 | Zbl 1029.60001

[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 2344310 | Zbl 1136.82391

[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 832433 | Zbl 0493.28007

[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 1440141 | Zbl 0883.60022

[8] J. Doob. Stochastic Processes. Wiley Classics Library Edition. Wiley, New York, 1990. | MR 1038526 | Zbl 0696.60003

[9] A. Garsia and J. Lamperti. A discrete renewal theorem with infinite mean. Comment. Math. Helv. 37 (1962/1963) 221-234. | MR 148121 | Zbl 0114.08803

[10] G. Giacomin. Random Polymer Models. Imperial College Press, London, 2007. | MR 2380992 | Zbl 1125.82001

[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 2761989 | Zbl 1194.82042

[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 2726088 | Zbl 1221.82058

[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 2209438 | Zbl 1099.60004

[16] F. Spitzer. Principles of Random Walks, 2nd edition. Grad. Texts in Math. 34. Springer, New York, 1976. | MR 388547 | Zbl 0359.60003

[17] J. M. Steele. Kingman's subadditive ergodic theorem. Ann. Inst. Henri Poincaré Probab. Stat. 25 (1989) 93-98. | Numdam | MR 995293 | Zbl 0669.60039

[18] F. L. Toninelli. A replica-coupling approach to disordered pinning models. Comm. Math. Phys. 280 (2008) 389-401. | MR 2395475 | Zbl 1207.82026

[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 2581605 | Zbl 1180.82241