The Annals of Applied Probability

A renewal theory approach to periodic copolymers with adsorption

Francesco Caravenna, Giambattista Giacomin, and Lorenzo Zambotti

Full-text: Open access


We consider a general model of a heterogeneous polymer chain fluctuating in the proximity of an interface between two selective solvents. The heterogeneous character of the model comes from the fact that the monomer units interact with the solvents and with the interface according to some charges that they carry. The charges repeat themselves along the chain in a periodic fashion. The main question concerning this model is whether the polymer remains tightly close to the interface, a phenomenon called localization, or whether there is a marked preference for one of the two solvents, thus yielding a delocalization phenomenon.

In this paper, we present an approach that yields sharp estimates for the partition function of the model in all regimes (localized, delocalized and critical). This, in turn, makes possible a precise pathwise description of the polymer measure, obtaining the full scaling limits of the model. A key point is the closeness of the polymer measure to suitable Markov renewal processes, Markov renewal theory being one of the central mathematical tools of our analysis.

Article information

Ann. Appl. Probab., Volume 17, Number 4 (2007), 1362-1398.

First available in Project Euclid: 10 August 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Random walks renewal theory Markov renewal theory scaling limits polymer models wetting models


Caravenna, Francesco; Giacomin, Giambattista; Zambotti, Lorenzo. A renewal theory approach to periodic copolymers with adsorption. Ann. Appl. Probab. 17 (2007), no. 4, 1362--1398. doi:10.1214/105051607000000159.

Export citation


  • Alexander, K. S. and Sidoravicius, V. (2006). Pinning of polymers and interfaces by random potentials. Ann. Appl. Probab. 16 636–669.
  • Asmussen, S. (2003). Applied Probability and Queues, 2nd ed. Springer, New York.
  • Bolthausen, E. (1976). On a functional central limit theorem for random walks conditioned to stay positive. Ann. Probab. 4 480–485.
  • Bolthausen, E. and den Hollander, F. (1997). Localization transition for a polymer near an interface. Ann. Probab. 25 1334–1366.
  • Bolthausen, E. and Giacomin, G. (2005). Periodic copolymers at selective interfaces: A large deviations approach. Ann. Appl. Probab. 15 963–983.
  • Caravenna, F., Giacomin, G. and Zambotti, L. (2006). Sharp asymptotic behavior for wetting models in $(1+1)$-dimension. Electron. J. Probab. 11 345–362.
  • Caravenna, F., Giacomin, G. and Zambotti, L. (2007). Infinite volume limits of polymer chains with periodic charges. Markov Process. Related Fields. To appear. Available at
  • Caravenna, F., Giacomin, G. and Zambotti, L. (2007). Tightness conditions for polymer measures. Available at
  • Deuschel, J.-D. Giacomin, G. and Zambotti, L. (2005). Scaling limits of equilibrium wetting models in $(1+1)$-dimension. Probab. Theory Related Fields 132 471–500.
  • Doney, R. A. (1997). One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Related Fields 107 451–465.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications II, 2nd ed. Wiley, New York.
  • Fitzsimmons, P. J., Fristedt, B. and Maisonneuve, B. (1985). Intersections and limits of regenerative sets. Z. Wahrsch. Verw. Gebiete 70 157–173.
  • Galluccio, S. and Graber, R. (1996). Depinning transition of a directed polymer by a periodic potential: A $d$-dimensional solution. Phys. Rev. E 53 R5584–R5587.
  • Garsia, A. and Lamperti, J. (1963). A discrete renewal theorem with infinite mean. Comment. Math. Helv. 37 221–234.
  • Giacomin, G. (2007). Random Polymer Models. Imperial College Press, London.
  • Giacomin, G. and Toninelli, F. L. (2005). Estimates on path delocalization for copolymers at selective interfaces. Probab. Theory Related Fields 133 464–482.
  • Isozaki, Y. and Yoshida, N. (2001). Weakly pinned random walk on the wall: Pathwise descriptions of the phase transition. Stochastic Process. Appl. 96 261–284.
  • Kaigh, W. D. (1976). An invariance principle for random walk conditioned by a late return to zero. Ann. Probab. 4 115–121.
  • Kingman, J. F. C. (1961). A convexity property of positive matrices. Quart. J. Math. Oxford Ser. (2) 12 283–284.
  • Le Gall, J.-F. and Rosen, J. (1991). The range of stable random walks. Ann. Probab. 19 650–705.
  • Matheron, G. (1975). Random Sets and Integral Geometry. Wiley, New York.
  • Minc, H. (1988). Nonnegative Matrices. Wiley, New York.
  • Monthus, C., Garel, T. and Orland, H. (2000). Copolymer at a selective interface and two dimensional wetting: A grand canonical approach. Eur. Phys. J. B 17 121–130.
  • Naidedov, A. and Nechaev, S. (2001). Adsorption of a random heteropolymer at a potential well revisited: Location of transition point and design of sequences. J. Phys. A: Math. Gen. 34 5625–5634.
  • Nechaev, S. and Zhang, Y.-C. (1995). Exact solution of the 2D wetting problem in a periodic potential. Phys. Rev. Lett. 74 1815–1818.
  • Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Springer, Berlin.
  • Sommer, J.-U. and Daoud, M. (1995). Copolymers at selective interfaces. Europhys. Lett. 32 407–412.
  • Soteros, C. E. and Whittington, S. G. (2004). The statistical mechanics of random copolymers. J. Phys. A: Math. Gen. 37 R279–R325.