Copolymer at selective interfaces and pinning potentials : weak coupling limits
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 1, p. 175-200

We consider a simple random walk of length N, denoted by ( S i ) i { 1 , l d o t s , N } , and we define ( w i ) i 1 a sequence of centered i.i.d. random variables. For K we define ( ( γ i - K , ... , γ i K ) ) i 1 an i.i.d sequence of random vectors. We set β , λ 0 and h 0 , and transform the measure on the set of random walk trajectories with the hamiltonian λ i = 1 N ( w i + h ) sign ( S i ) + β j = - K K i = 1 N γ i j 1 { S i = j } . This transformed path measure describes an hydrophobic(philic) copolymer interacting with a layer of width 2 K 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.

On considère une marche aléatoire simple de taille N, que l'on note ( S i ) i { 1 , l d o t s , N } , et on définit ( w i ) i 1 une suite de variables aléatoires i.i.d. et centrées. Pour tous K on définit ( ( γ i - K , ... , γ i K ) ) i 1 une suite de vecteurs aléatoires i.i.d. On pose β , λ 0 et h 0 , et on transforme la mesure de l'ensemble des trajectoires de la marche aléatoire avec le hamiltonien λ i = 1 N ( w i + h ) sign ( S i ) + β j = - K K i = 1 N γ i j 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 λ , h et β 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.

DOI : https://doi.org/10.1214/07-AIHP160
Classification:  82B41,  60K35,  60K37
Keywords: polymers, localization-delocalization transition, pinning, random walk, weak coupling
@article{AIHPB_2009__45_1_175_0,
     author = {Petrelis, Nicolas},
     title = {Copolymer at selective interfaces and pinning potentials : weak coupling limits},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {1},
     year = {2009},
     pages = {175-200},
     doi = {10.1214/07-AIHP160},
     zbl = {1172.82318},
     mrnumber = {2500234},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2009__45_1_175_0}
}
Petrelis, Nicolas. Copolymer at selective interfaces and pinning potentials : weak coupling limits. Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 1, pp. 175-200. doi : 10.1214/07-AIHP160. http://www.numdam.org/item/AIHPB_2009__45_1_175_0/

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