The Annals of Probability

The weak coupling limit of disordered copolymer models

Francesco Caravenna and Giambattista Giacomin

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A copolymer is a chain of repetitive units (monomers) that are almost identical, but they differ in their degree of affinity for certain solvents. This difference leads to striking phenomena when the polymer fluctuates in a nonhomogeneous medium, for example, made of two solvents separated by an interface. One may observe, for instance, the localization of the polymer at the interface between the two solvents. A discrete model of such system, based on the simple symmetric random walk on ℤ, has been investigated in [8], notably in the weak polymer-solvent coupling limit, where the convergence of the discrete model toward a continuum model, based on Brownian motion, has been established. This result is remarkable because it strongly suggests a universal feature of copolymer models. In this work, we prove that this is indeed the case. More precisely, we determine the weak coupling limit for a general class of discrete copolymer models, obtaining as limits a one-parameter [α ∈ (0, 1)] family of continuum models, based on α-stable regenerative sets.

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Ann. Probab., Volume 38, Number 6 (2010), 2322-2378.

First available in Project Euclid: 24 September 2010

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Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 60K37: Processes in random environments 60K05: Renewal theory 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Copolymer renewal process regenerative set phase transition coarse-graining weak coupling limit universality


Caravenna, Francesco; Giacomin, Giambattista. The weak coupling limit of disordered copolymer models. Ann. Probab. 38 (2010), no. 6, 2322--2378. doi:10.1214/10-AOP546.

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