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

We consider the free energy $F(β)$ of the directed polymers in random environment in $1+1$-dimension. It is known that $F(β)$ is of order $-β^4$ as $β\to 0$. In this paper, we will prove that under a certain condition of the potential, \begin{align*} \lim_{β\to 0}\frac{F(β)}{β^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 $\{\mathcal{Z}_β(t,x):t\geq 0,x\in\mathbb{R}\}$ is the unique mild solution to the stochastic heat equation \begin{align*} \frac{\partial}{\partial t}\mathcal{Z}=\frac{1}{2}Δ\mathcal{Z}+β\mathcal{Z}{\dot{\mathcal W}},\ \ \lim_{t\to 0}\mathcal{Z}(t,x)dx=δ_{0}(dx), \end{align*} where $\mathcal{W}$ is a time-space white noise and \begin{align*} \mathcal{Z}_β(t)=\int_\mathbb{R}\mathcal{Z}_β(t,x)dx. \end{align*}

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