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(-c(epsilon)/beta(2+epsilon)) for small beta, where c(epsilon) > 0 is a constant depending on epsilon > 0. Also, we will suggest a strategy to get a sharper asymptotics. We denote by (Omega, F, P) a probability space. (C) 2014 AIP Publishing LLC.