Copolymer at selective interfaces and pinning potentials: weak coupling limits

2006/09/28 by Nicolas Pétrélis, Nicolas Petrelis, Petrelis, Nicolas
Materials Science · Mathematics · Physics and Astronomy · #Block Copolymer Self-Assembly #FOS: Mathematics #Force Microscopy Techniques and Applications #Material Dynamics and Properties #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.math/0609814

26 pages

openalex publication_date 2006/09/28 · arxiv created 2007/11/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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) \sign(S_i)+β∑_j=-K^K∑_i=1^N γ_i^j \boldsymbol1_\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 \citeBDH. Our result shows, in particular, that the randomness of the pinning around the interface vanishes as the coupling becomes weaker.

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