无序共聚物模型的弱耦合极限
The weak coupling limit of disordered copolymer models
摘要：
共聚物是几乎相同的重复单元（单体）的链，但它们对某些溶剂的亲和力不同。当聚合物在非均匀介质中波动时，这种差异会导致惊人的现象，例如，由界面分离的两种溶剂组成。例如，可以观察到聚合物在两种溶剂之间的界面处的定位。基于$\mathbb {Z}$上的简单对称随机游动的这种系统的离散模型已经在[Bolthausen和den Hollander，Ann. Probab. 25（1997），1334-1366]，特别是在弱聚合物-溶剂耦合极限中，其中已经建立了基于布朗运动的离散模型向连续模型的收敛。这个结果是显著的，因为它强烈地暗示了共聚物模型的普遍特征。在这项工作中，我们证明了情况确实如此。更准确地说，我们确定了弱耦合限制一般类的离散共聚物模型，作为限制获得一个参数[ $α\in(0,1)$ ]家庭的连续模型，基于$α$ -稳定再生集。
英文摘要：
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 [$α\in(0,1)$] family of continuum models, based on $α$-stable regenerative sets.
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