  We consider first passage percolation with i.i.d. weights on edges of the
d-dimensional cubic lattice. Under the assumptions that a weight is equal to
zero with probability smaller than the critical probability of bond percolation
in the d-dimensional cubic lattice, and has moments bigger than 1, we
investigate upper bounds on the so-called non-random fluctuations of the model.
In addition, we give an application of our result to a lower bound for variance
of the first passage percolation in the case where the limit shape has flat
edges.
