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 -> 0 [3, 28, 42]. In this paper, we will prove that under a certain dimension free concentration condition on the potential, lim(beta -> 0) F(beta)/beta(4 )= lim(T ->infinity )1/T P-Z( )[logZ (root 2)(T)]( )=-1/6, where {Z(beta)(t, x) : t >= 0, x is an element of R} is the unique mild solution to the stochastic heat equation partial derivative/partial derivative t Z = 1/2 Delta Z +beta Z(W) over dot, lim(t -> 0) Z(t,x)dx =delta(0)(dx), where W is a time-space white noise and Z(beta)(t) = integral(R) Z(beta)(t, x)dx.