  We calculate exactly the velocity and diffusion constant of a microscopic
stochastic model of $N$ evolving particles which can be described by a noisy
traveling wave equation with a noise of order $N^{-1/2}$. Our model can be
viewed as the infinite range limit of a directed polymer in random medium with
$N$ sites in the transverse direction. Despite some peculiarities of the
traveling wave equations in the absence of noise, our exact solution allows us
to test the validity of a simple cutoff approximation and to show that, in the
weak noise limit, the position of the front can be completely described by the
effect of the noise on the first particle.
