Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Copolymer at selective interfaces and pinning potentials: Weak coupling limits

Nicolas Petrelis

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Abstract

We consider a simple random walk of length $N$, denoted by $(S_i)_{i∈\{1, …, N\}}$, and we define $(w_i)_{i≥1}$ a sequence of centered i.i.d. random variables. For $K∈ℕ$ we define $((γ_i^{−K}, …, γ_i^K))_{i≥1}$ 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 $λ∑_{i=1}^N(w_i+h)\mathrm{sign}(S_i)+β∑_{j=−K}^K∑_{i=1}^Nγ_i^j\mathbf{1}_{\{S_i=j\}}$. 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 $λ$, $h$ 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.

Résumé

On considère une marche aléatoire simple de taille $N$, que l’on note $(S_i)_{i∈\{1, …, N\}}$, et on définit $(w_i)_{i≥1}$ une suite de variables aléatoires i.i.d. et centrées. Pour tous $K∈ℕ\cup\{0\}$ on définit $((γ_i^{−K}, …, γ_i^K))_{i≥1}$ une suite de vecteurs aléatoires i.i.d. On pose $β∈ℝ$, $λ≥0$ et $h≥0$, et on transforme la mesure de l’ensemble des trajectoires de la marche aléatoire avec le hamiltonien $λ∑_{i=1}^N(w_i+h)\mathrm{sign}(S_i)+β∑_{j=−K}^K∑_{i=1}^Nγ_i^j\mathbf{1}_{\{S_i=j\}}$. 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 $λ$, $h$ 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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 45, Number 1 (2009), 175-200.

Dates
First available in Project Euclid: 12 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1234469977

Digital Object Identifier
doi:10.1214/07-AIHP160

Mathematical Reviews number (MathSciNet)
MR2500234

Zentralblatt MATH identifier
1172.82318

Subjects
Primary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments

Keywords
Polymers Localization-delocalization transition Pinning Random walk Weak coupling

Citation

Petrelis, Nicolas. Copolymer at selective interfaces and pinning potentials: Weak coupling limits. Ann. Inst. H. Poincaré Probab. Statist. 45 (2009), no. 1, 175--200. doi:10.1214/07-AIHP160. https://projecteuclid.org/euclid.aihp/1234469977


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