Copolymer at selective interfaces and pinning potentials : weak coupling limits

Nicolas Petrelis

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 1, page 175-200
  • ISSN: 0246-0203

Abstract

top
We consider a simple random walk of length N, denoted by (Si)i∈{1, …, N}, and we define (wi)i≥1 a sequence of centered i.i.d. random variables. For K∈ℕ we define ((γi−K, …, γiK))i≥1 an i.i.d sequence of random vectors. We set β∈ℝ, λ≥0 and h≥0, and transform the measure on the set of random walk trajectories with the hamiltonian λ∑i=1N(wi+h)sign(Si)+β∑j=−KK∑i=1Nγij1{Si=j}. This transformed path measure describes an hydrophobic(philic) copolymer interacting with a layer of width 2K around an interface between oil and water. In the present article we prove the convergence in the limit of weak coupling (when λ, h and β tend to 0) of this discrete model towards its continuous counterpart. To that aim we further develop a technique of coarse graining introduced by Bolthausen and den Hollander in Ann. Probab.25 (1997) 1334–1366. Our result shows, in particular, that the randomness of the pinning around the interface vanishes as the coupling becomes weaker.

How to cite

top

Petrelis, Nicolas. "Copolymer at selective interfaces and pinning potentials : weak coupling limits." Annales de l'I.H.P. Probabilités et statistiques 45.1 (2009): 175-200. <http://eudml.org/doc/78014>.

@article{Petrelis2009,
abstract = {We consider a simple random walk of length N, denoted by (Si)i∈\{1, …, N\}, and we define (wi)i≥1 a sequence of centered i.i.d. random variables. For K∈ℕ we define ((γi−K, …, γiK))i≥1 an i.i.d sequence of random vectors. We set β∈ℝ, λ≥0 and h≥0, and transform the measure on the set of random walk trajectories with the hamiltonian λ∑i=1N(wi+h)sign(Si)+β∑j=−KK∑i=1Nγij1\{Si=j\}. This transformed path measure describes an hydrophobic(philic) copolymer interacting with a layer of width 2K around an interface between oil and water. In the present article we prove the convergence in the limit of weak coupling (when λ, h and β tend to 0) of this discrete model towards its continuous counterpart. To that aim we further develop a technique of coarse graining introduced by Bolthausen and den Hollander in Ann. Probab.25 (1997) 1334–1366. Our result shows, in particular, that the randomness of the pinning around the interface vanishes as the coupling becomes weaker.},
author = {Petrelis, Nicolas},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {polymers; localization-delocalization transition; pinning; random walk; weak coupling},
language = {eng},
number = {1},
pages = {175-200},
publisher = {Gauthier-Villars},
title = {Copolymer at selective interfaces and pinning potentials : weak coupling limits},
url = {http://eudml.org/doc/78014},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Petrelis, Nicolas
TI - Copolymer at selective interfaces and pinning potentials : weak coupling limits
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 1
SP - 175
EP - 200
AB - We consider a simple random walk of length N, denoted by (Si)i∈{1, …, N}, and we define (wi)i≥1 a sequence of centered i.i.d. random variables. For K∈ℕ we define ((γi−K, …, γiK))i≥1 an i.i.d sequence of random vectors. We set β∈ℝ, λ≥0 and h≥0, and transform the measure on the set of random walk trajectories with the hamiltonian λ∑i=1N(wi+h)sign(Si)+β∑j=−KK∑i=1Nγij1{Si=j}. This transformed path measure describes an hydrophobic(philic) copolymer interacting with a layer of width 2K around an interface between oil and water. In the present article we prove the convergence in the limit of weak coupling (when λ, h and β tend to 0) of this discrete model towards its continuous counterpart. To that aim we further develop a technique of coarse graining introduced by Bolthausen and den Hollander in Ann. Probab.25 (1997) 1334–1366. Our result shows, in particular, that the randomness of the pinning around the interface vanishes as the coupling becomes weaker.
LA - eng
KW - polymers; localization-delocalization transition; pinning; random walk; weak coupling
UR - http://eudml.org/doc/78014
ER -

References

top
  1. [1] S. Albeverio and X. Y. Zhou. Free energy and some sample path properties of a random walk with random potential. J. Statist. Phys. 83 (1996) 573–622. Zbl1081.82559MR1386352
  2. [2] K. S. Alexander. The effect of disorder on polymer depinning transitions. Commun. Math. Phys. 279 (2008) 117–146. Zbl1175.82034MR2377630
  3. [3] K. S. Alexander and V. Sidoravicius. Pinning of polymers and interfaces by random potentials. Ann. Appl. Probab. 16 (2006) 636–669. Zbl1145.82010MR2244428
  4. [4] T. Bodineau and G. Giacomin. On the localization transition of random copolymers near selective interfaces. J. Statist. Phys. 117 (2004) 801–818. Zbl1089.82031MR2107896
  5. [5] M. Biskup and F. den Hollander. A heteropolymer near a linear interface. Ann. Appl. Prob. 25 (1999) 668–876. Zbl0971.60098MR1722277
  6. [6] E. Bolthausen and F. den Hollander. Localization for a polymer near an interface. Ann. Probab. 25 (1997) 1334–1366. Zbl0885.60022MR1457622
  7. [7] F. Caravenna, G. Giacomin and M. Gubinelli. A numerical approach to copolymer at selective interfaces. J. Statsit. Phys. 122 (2006) 799–832. Zbl1149.82357MR2213950
  8. [8] B. Derrida, V. Hakim and J. Vannimenus. Effect of disorder on two-dimensional wetting. J. Statist. Phys. 66 (1992) 1189–1213. Zbl0900.82051MR1156401
  9. [9] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. II. Wiley, New York (1971). Zbl0219.60003MR270403
  10. [10] G. Giacomin. Localization phenomena in random polymer models. Note for the course in Pisa and in the graduate school of Paris 6, 2003. http://www.proba.jussieu.fr/pageperso/giacomin/pub/publicat.html. 
  11. [11] G. Giacomin. Random Polymer Models. Imperial College Press, London, 2007. Zbl1125.82001MR2380992
  12. [12] G. Giacomin and F. L. Toninelli. Estimates on path delocalization for copolymers at selective interfaces. Probab. Theory Related Fields 133 (2005) 464–482. Zbl1098.60089MR2197110
  13. [13] G. Giacomin and F. L. Toninelli. The localized phase of a disordered copolymer with adsorption. Alea 1 (2006) 149–180. Zbl1134.82006MR2249653
  14. [14] E. W. James, C. E. Soteros and S. G. Whittington. Localization of a random copolymer at an interface: an exact enumeration study. J. Phys. A 36 (2003) 11575–11584. Zbl1039.82016MR2025861
  15. [15] E. Janvresse, T. de la Rue and Y. Velenik. Pinning by a sparse potential. Stochastic. Process. Appl. 115 (2005) 1323–1331. Zbl1079.60077MR2152377
  16. [16] I. Karatzas and S. E. Shreeve. Brownian Motion and Stochastic Calculus. Springer, New York, 1991. Zbl0734.60060MR1121940
  17. [17] N. Pétrélis. Polymer pinning at an interface. Stoch. Proc. Appl. 116 (2006) 1600–1621. Zbl1129.82016MR2269218
  18. [18] N. Pétrélis. Thesis, University of Rouen, France. Online thesis, 2006. 
  19. [19] P. Révész. Local Time and Invariance. Springer, Berlin, 1981. Zbl0456.60029MR655268
  20. [20] D. Revuz and M. Yor. Continuous Martingales and Brownian Motions. Wiley, New York, 1992. MR1083357
  21. [21] Q.-M. Shao. Strong approximation theorems for independent variables and their applications. J. Multivariate Anal. 52 107–130. Zbl0817.60027MR1325373
  22. [22] C. E. Soteros and S. G. Whittington. The statistical mechanics of random copolymers. J. Phys. A: Math. Gen. 37 (2004) R279–R325. Zbl1073.82015MR2097625
  23. [23] Y. G. Sinai. A random walk with a random potential. Theory Probab. Appl. 38 (1993) 382–385. Zbl0807.60069MR1317991

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.