Free energy of directed polymers in random environment in $1+1$-dimension at high temperature

We consider the free energy $F(\beta )$ of the directed polymers in random environment in $1+1$-dimension. It is known that $F(\beta )$ is of order $-{\beta }^4$ as $\beta \to 0$. In this paper, we will prove that under a certain condition of the potential, \begin{align*} \lim_{\beta\to 0}\frac{F(\beta)}{\beta^4}=\lim_{T\to\infty}\frac{1}{T}P_{$\mathcal{Z}$}\left[\log $\mathcal{Z}${\sqrt{2}}(T)\right] =-\frac{1}{6}, \end{align*} where $\{{\mathcal{Z}}_{\beta }(t,x):t\ge 0,x\in \mathbb{R}\}$ is the unique mild solution to the stochastic heat equation \begin{align*} \frac{\partial}{\partial t}$\mathcal{Z}$=\frac{1}{2}\Delta $\mathcal{Z}$+\beta $\mathcal{Z}${\dot{\mathcal W}},\ \ \lim{t\to 0}$\mathcal{Z}$(t,x)dx=\delta_{0}(dx), \end{align*} where $\mathcal{W}$ is a time-space white noise and \begin{align*} $\mathcal{Z}$\beta(t)=\int$\mathbb{R}$$\mathcal{Z}$_\beta(t,x)dx. \end{align*}