Two-temperature statistics of free energies in (1+1) directed polymers

TL;DR

This paper investigates the joint statistical behavior of free energies at two different temperatures in (1+1) directed polymers, revealing scaling laws and tail asymptotics linked to Tracy-Widom distribution.

Contribution

It derives the temperature dependence of free energy differences and connects their tail behavior to Tracy-Widom distribution in directed polymers.

Findings

Fluctuations scale as (1 - T1/T2)^{1/3} for close temperatures.

Left tail of free energy difference matches Tracy-Widom tail.

Provides a scaling relation for free energy differences at different temperatures.

Abstract

The joint statistical properties of two free energies computed at two different temperatures in {\it the same sample} of $(1+1)$ directed polymers is studied in terms of the replica technique. The scaling dependence of the reduced free energies difference ${\cal F} = F(T_{1})/T_{1} - F(T_{2})/T_{2}$ on the two temperatures $T_{1}$ and $T_{2}$ is derived. In particular, it is shown that if the two temperatures $T_{1} \, < \, T_{2}$ are close to each other the typical value of the fluctuating part of the reduced free energies difference ${\cal F}$ is proportional to $(1 - T_{1}/T_{2})^{1/3}$. It is also shown that the left tail asymptotics of this free energy difference probability distribution function coincides with the corresponding tail of the TW distribution.