A remark on the bound for the free energy of directed polymers in random environment in 1+2 dimension

TL;DR

This paper investigates the upper bound of the free energy of directed polymers in a 1+2 dimensional random environment, providing a specific exponential bound for small inverse temperature and proposing a method for sharper asymptotics.

Contribution

It establishes a new upper bound for the free energy in 1+2 dimensions and suggests a strategy for obtaining more precise asymptotic estimates.

Findings

Proves an exponential upper bound for the free energy at small 

Shows the free energy is strictly negative for  not zero

Proposes a strategy for sharper asymptotic analysis

Abstract

We consider the behavior of the quantity $p(\beta)$; the free energy of directed polymers in random environment in $1+2$ dimension, where $\beta$ is inverse temperature. We know that the free energy is strictly negative when $\beta$ is not zero. In this paper, we will prove that $p(\beta)$ is bounded from above by $-\exp\left( -\frac{c_\varepsilon}{\beta^{2+\varepsilon}} \right)$ for small $\beta$, where $c_\varepsilon>0$ is a constant depending on $\varepsilon>0$. Also, we will suggest a strategy to get a sharper asymptotics.