A numerical approach to copolymers at selective interfaces

TL;DR

This paper combines numerical and rigorous methods to better understand the phase transition of a copolymer model at a selective interface, providing insights into the critical line and the behavior of the polymer.

Contribution

It introduces a rigorous statistical test for phase classification, refines bounds on the critical line, and analyzes the asymptotic behavior of the partition function.

Findings

Critical line lies strictly between known bounds.

Lower bound on the critical line is confirmed not to coincide with the actual transition.

Polymer behavior in the delocalized phase involves rare disorder stretches and may exhibit Brownian scaling.

Abstract

We consider a model of a random copolymer at a selective interface which undergoes a localization/delocalization transition. In spite of the several rigorous results available for this model, the theoretical characterization of the phase transition has remained elusive and there is still no agreement about several important issues, for example the behavior of the polymer near the phase transition line. From a rigorous viewpoint non coinciding upper and lower bounds on the critical line are known. In this paper we combine numerical computations with rigorous arguments to get to a better understanding of the phase diagram. Our main results include: - Various numerical observations that suggest that the critical line lies strictly in between the two bounds. - A rigorous statistical test based on concentration inequalities and super-additivity, for determining whether a given point of the phase diagram is in the localized phase. This is applied in particular to show that, with a very low level of error, the lower bound does not coincide with the critical line. - An analysis of the precise asymptotic behavior of the partition function in the delocalized phase, with particular attention to the effect of rare atypical stretches in the disorder sequence and on whether or not in the delocalized regime the polymer path has a Brownian scaling. - A new proof of the lower bound on the critical line. This proof relies on a characterization of the localized regime which is more appealing for interpreting the numerical data.