Anomalous Fluctuations of Directed Polymers in Random Media

TL;DR

This paper analyzes large-scale fluctuations in directed polymers within random media, revealing power-law distributions of rare regions that dominate the physics, using a mapping to the noisy-Burgers' equation and symmetry considerations.

Contribution

It provides a systematic analysis of fluctuations in directed polymers, linking rare region statistics to symmetry breaking and Goldstone modes, advancing understanding of glassy phases.

Findings

Power law tail in the size distribution of rare regions.

Rare regions dominate the physics of the low temperature phase.

Continuous tilt symmetry leads to large active excitations.

Abstract

A systematic analysis of large scale fluctuations in the low temperature pinned phase of a directed polymer in a random potential is described. These fluctuations come from rare regions with nearly degenerate ``ground states''. The probability distribution of their sizes is found to have a power law tail. The rare regions in the tail dominate much of the physics. The analysis presented here takes advantage of the mapping to the noisy-Burgers' equation. It complements a phenomenological description of glassy phases based on a scaling picture of droplet excitations and a recent variational approach with ``broken replica symmetry''. It is argued that the power law distribution of large thermally active excitations is a consequence of the continuous statistical ``tilt'' symmetry of the directed polymer, the breaking of which gives rise to the large active excitations in a manner analogous to the appearance of Goldstone modes in pure systems with a broken continuous symmetry.