A remark on the bound for the free energy of directed polymers in random   environment in 1+2 dimension
  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. 
Makoto Nakashima
math.PR