Copolymer at selective interfaces and pinning potentials : weak coupling limits

Petrelis, Nicolas

Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 1, pp. 175-200.

On considère une marche aléatoire simple de taille $N$, que l'on note ${\left(S_i\right)}_{i\in \{1,ldots,N\}}$, et on définit ${\left(w_i\right)}_{i\ge 1}$ une suite de variables aléatoires i.i.d. et centrées. Pour tous $K\in \mathbb{N}$ on définit ${\left(({\gamma }_i^{-K},...,{\gamma }_i^K)\right)}_{i\ge 1}$ une suite de vecteurs aléatoires i.i.d. On pose $\beta \in ℝ$, $\lambda \ge 0$ et $h\ge 0$, et on transforme la mesure de l'ensemble des trajectoires de la marche aléatoire avec le hamiltonien $\lambda {\sum }_{i=1}^N(w_i+h)\mathrm{sign}\left(S_i\right)+\beta {\sum }_{j=-K}^K{\sum }_{i=1}^N{\gamma }_i^j{\mathbf{1}}_{\{S_i=j\}}$. Cette mesure perturbée décrit un copolymère hydrophobe(phile) en interaction avec une bande de taille 2K autour d'une interface huile-eau. Dans cette article nous prouvons la convergence dans la limite d'un couplage faible (quand $\lambda$, $h$ et $\beta$ tendent vers 0) de ce modèle discret vers son homologue continu. Dans ce but, nous développons une technique de coarse graining introduite par Bolthausen et den Hollander dans Ann. Probab. 25 (1997) 1334-1366. Ce résultat montre en particulier que le caractère aléatoire de l'accrochage autour de l'interface disparaît à mesure que le couplage s'affaiblit.

We consider a simple random walk of length $N$, denoted by ${\left(S_i\right)}_{i\in \{1,ldots,N\}}$, and we define ${\left(w_i\right)}_{i\ge 1}$ a sequence of centered i.i.d. random variables. For $K\in \mathbb{N}$ we define ${\left(({\gamma }_i^{-K},...,{\gamma }_i^K)\right)}_{i\ge 1}$ an i.i.d sequence of random vectors. We set $\beta \in ℝ$, $\lambda \ge 0$ and $h\ge 0$, and transform the measure on the set of random walk trajectories with the hamiltonian $\lambda {\sum }_{i=1}^N(w_i+h)\mathrm{sign}\left(S_i\right)+\beta {\sum }_{j=-K}^K{\sum }_{i=1}^N{\gamma }_i^j{\mathbf{1}}_{\{S_i=j\}}$. 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 $\lambda$, $h$ and $\beta$ 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.

[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. | MR | Zbl

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

[3] K. S. Alexander and V. Sidoravicius. Pinning of polymers and interfaces by random potentials. Ann. Appl. Probab. 16 (2006) 636-669. | MR | Zbl

[4] T. Bodineau and G. Giacomin. On the localization transition of random copolymers near selective interfaces. J. Statist. Phys. 117 (2004) 801-818. | MR | Zbl

[5] M. Biskup and F. Den Hollander. A heteropolymer near a linear interface. Ann. Appl. Prob. 25 (1999) 668-876. | MR | Zbl

[6] E. Bolthausen and F. Den Hollander. Localization for a polymer near an interface. Ann. Probab. 25 (1997) 1334-1366. | MR | Zbl

[7] F. Caravenna, G. Giacomin and M. Gubinelli. A numerical approach to copolymer at selective interfaces. J. Statsit. Phys. 122 (2006) 799-832. | MR | Zbl

[8] B. Derrida, V. Hakim and J. Vannimenus. Effect of disorder on two-dimensional wetting. J. Statist. Phys. 66 (1992) 1189-1213. | MR | Zbl

[9] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. II. Wiley, New York (1971). | MR | Zbl

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

[12] G. Giacomin and F. L. Toninelli. Estimates on path delocalization for copolymers at selective interfaces. Probab. Theory Related Fields 133 (2005) 464-482. | MR | Zbl

[13] G. Giacomin and F. L. Toninelli. The localized phase of a disordered copolymer with adsorption. Alea 1 (2006) 149-180. | MR | Zbl

[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. | MR | Zbl

[15] E. Janvresse, T. De La Rue and Y. Velenik. Pinning by a sparse potential. Stochastic. Process. Appl. 115 (2005) 1323-1331. | MR | Zbl

[16] I. Karatzas and S. E. Shreeve. Brownian Motion and Stochastic Calculus. Springer, New York, 1991. | MR | Zbl

[17] N. Pétrélis. Polymer pinning at an interface. Stoch. Proc. Appl. 116 (2006) 1600-1621. | MR | Zbl

[18] N. Pétrélis. Thesis, University of Rouen, France. Online thesis, 2006.

[19] P. Révész. Local Time and Invariance. Springer, Berlin, 1981. | MR | Zbl

[20] D. Revuz and M. Yor. Continuous Martingales and Brownian Motions. Wiley, New York, 1992. | MR

[21] Q.-M. Shao. Strong approximation theorems for independent variables and their applications. J. Multivariate Anal. 52 107-130. | MR | Zbl

[22] C. E. Soteros and S. G. Whittington. The statistical mechanics of random copolymers. J. Phys. A: Math. Gen. 37 (2004) R279-R325. | MR | Zbl

[23] Y. G. Sinai. A random walk with a random potential. Theory Probab. Appl. 38 (1993) 382-385. | MR | Zbl