Directed polymers in a random medium: a variational approach

TL;DR

This paper applies a Gaussian variational method to study directed polymers in random media, revealing different behaviors and critical exponents depending on the dimension, and identifying phase transitions and polymer segmentation phenomena.

Contribution

It introduces a disorder-dependent variational approach that captures the polymer's behavior across different dimensions, including phase transitions and replica symmetry breaking.

Findings

For d<2, two variational solutions with distinct critical exponents.

No phase transition for 2<d<4, but polymer segmentation occurs.

Phase transition at d>4 similar to Derrida's random energy models.

Abstract

A disorder-dependent Gaussian variational approach is applied to the problem of a $d$ dimensional polymer chain in a random medium (or potential). Two classes of variational solutions are obtained. For $d<2$, these two classes may be interpreted as domain and domain wall. The critical exponent $\nu$ describing the polymer width is $\nu={1\over (4-d)}$ (domain solution) or $\nu={3\over (d+4)}$ (domain wall solution). The domain wall solution is equivalent to the (full) replica symmetry breaking variational result. For $d>2$, we find $\nu={1\over 2}$. No evidence of a phase transition is found for $2< d< 4$: one of the variational solutions suggests that the polymer chain breaks into Imry-Ma segments, whose probability distribution is calculated. For $d>4$, the other variational solution undergoes a phase transition, which has some similarity with B. Derrida's random energy models.