Scaling Limits and Universality in Disordered Copolimer Models
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Mon, 01/02/2010 14:15 |
Giambattista Giamcomin (University of Paris Diderot) |
Stochastic Analysis Seminar  |
Eagle House |
| 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 non-homogeneous medium, for example made up by two solvents separated by an interface. One may observe, for exmple, the localization of the polymer at the interface between the two solvents. Much of the literature on the subject focuses on the most basic model based on the simple symmetric random walk on the integers, but E. Bolthausen and F. den Hollander (AP 1997) pointed out the convergence of the (rescaled) free energy of such a discrete model toward the free energy of a continuum model, based on Brownian motion, in the limit of weak polymer-solvent coupling. This result is remarkable because it strongly suggests a universal feature for 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. | |||