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

Abstract

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], \[\lim_{\beta\to0}\frac{1}{\beta^4}F(\beta)=-\frac{1}{6}. \] In this paper, we obtain the same asymptotics without using the concentration inequality.