  In this paper, we obtain optimal uniform lower tail estimates for the
probability distribution of the properly scaled length of the longest up/right
path of the last passage site percolation model considered by Johansson in
[12]. The estimates are used to prove a lower tail moderate deviation result
for the model. The estimates also imply the convergence of moments, and also
provide a verification of the universal scaling law relating the longitudinal
and the transversal fluctuations of the model.
