A note on the asymptotics of the free energy of $1+1$ dimensional directed polymers in random environment at high temperature

The author gave the sharp asymptotic behavior of the free energy of $1+1$ dimensional directed polymers in random environment(DPRE) as the inverse temperature $\beta \to 0$ under the assumption that random environment satisfies a certain concentration inequality in [Nak19], $$\underset{\beta \to 0}{\lim }\frac{1}{{\beta }^4}F(\beta )=-\frac{1}{6}\mathrm{.}$$ In this paper, we obtain the same asymptotics without using the concentration inequality.