Finite-temperature and finite-time scaling of the directed polymer free-energy with respect to its geometrical fluctuations

TL;DR

This paper investigates how the geometrical and free-energy fluctuations of directed polymers in random environments scale at finite temperature and correlation length, supported by analytical and numerical evidence.

Contribution

It proposes a universal scaling law linking geometrical and free-energy fluctuations of directed polymers considering finite temperature and correlation length.

Findings

Scaling law between geometrical and free-energy fluctuations derived

Numerical simulations confirm the analytical predictions

Applications to liquid crystal experiments discussed

Abstract

We study the fluctuations of the directed polymer in 1+1 dimensions in a Gaussian random environment with a finite correlation length {\xi} and at finite temperature. We address the correspondence between the geometrical transverse fluctuations of the directed polymer, described by its roughness, and the fluctuations of its free-energy, characterized by its two-point correlator. Analytical arguments are provided in favor of a generic scaling law between those quantities, at finite time, non-vanishing {\xi} and explicit temperature dependence. Numerical results are in good agreement both for simulations on the discrete directed polymer and on a continuous directed polymer (with short-range correlated disorder). Applications to recent experiments on liquid crystals are discussed.