We consider the stochastic heat equation in one space dimension and compute
- for a particular choice of the initial datum - the exact long time asymp
totic. In the Carmona-Molchanov approach to intermittence in non stationary
random media this corresponds to the identification of the sample Lyapunov
exponent, Equivalently, by interpreting the solution as the partition func
tion of a directed polymer in a random environment, we obtain a weak law of
large numbers for the quenched free energy. The result agrees with the one
obtained in the physical literature via the replica method. The proof is b
ased on a representation of the solution in terms of the weakly asymmetric
exclusion process.