On quenched and annealed critical curves of random pinning model with finite range correlations. (English. French summary) Zbl 1276.82024

The paper is devoted to directed polymers pinned at a disordered and correlated interface. It is assumed that the disorder sequence is a \(q\)-order moving average. It is established 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. The author is able to produce explicit values of the annealed critical curve for \(q= 1\) and \(q=2\) and a weak disorder asymptotic in the general case. By means of the renewal theory approach of pinning, it is established that the constructed processes from the annealed model are particular Markov renewal processes. It is considered the intersection of two replicas of such process and using the method of second moment, the author proves a result of disorder irrelevance.


82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60K37 Processes in random environments
60K05 Renewal theory
Full Text: DOI arXiv Euclid


[1] K. S. Alexander. The effect of disorder on polymer depinning transitions. Comm. Math. Phys. 279 (2008) 117-146. · Zbl 1175.82034 · doi:10.1007/s00220-008-0425-5
[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. · 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. · Zbl 1136.82391 · doi:10.1214/105051607000000159
[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.
[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. · Zbl 0883.60022 · doi:10.1007/s004400050093
[8] J. Doob. Stochastic Processes. Wiley Classics Library Edition . Wiley, New York, 1990. · Zbl 0696.60003
[9] A. Garsia and J. Lamperti. A discrete renewal theorem with infinite mean. Comment. Math. Helv. 37 (1962/1963) 221-234. · Zbl 0114.08803 · doi:10.1007/BF02566974
[10] G. Giacomin. Random Polymer Models . Imperial College Press, London, 2007. · 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. · Zbl 1194.82042 · doi:10.1007/978-3-7643-9891-0_11
[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. · Zbl 1221.82058 · doi:10.1214/ECP.v15-1572
[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. · Zbl 1099.60004
[16] F. Spitzer. Principles of Random Walks , 2nd edition. Grad. Texts in Math. 34 . Springer, New York, 1976. · Zbl 0359.60003
[17] J. M. Steele. Kingman’s subadditive ergodic theorem. Ann. Inst. Henri Poincaré Probab. Stat. 25 (1989) 93-98. · Zbl 0669.60039
[18] F. L. Toninelli. A replica-coupling approach to disordered pinning models. Comm. Math. Phys. 280 (2008) 389-401. · Zbl 1207.82026 · doi:10.1007/s00220-008-0469-6
[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. · Zbl 1180.82241 · doi:10.1007/978-3-540-92796-9_3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.