The weak coupling limit of disordered copolymer models

TL;DR

This paper establishes the universal behavior of disordered copolymer models in the weak coupling limit, showing convergence from discrete to a family of continuum models based on stable processes.

Contribution

It generalizes previous results by proving the weak coupling limit for a broad class of discrete copolymer models, leading to a continuum limit involving stable regenerative sets.

Findings

Convergence of discrete models to continuum models in the weak coupling limit.

Identification of a family of continuum models parameterized by alpha.

Universal behavior across different copolymer models.

Abstract

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 $\mathbb {Z}$, has been investigated in [Bolthausen and den Hollander, Ann. Probab. 25 (1997), 1334-1366], 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 [$\alpha \in(0,1)$] family of continuum models, based on $\alpha$-stable regenerative sets.