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

Abstract

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