  This paper studies the first passage percolation (FPP) model: each edge in
the cubic lattice is assigned a random passage time, and consideration is given
to the behavior of the percolation region $B(t)$, which consists of those
vertices that can be reached from the origin within a time $t > 0$. Cox and
Durrett showed the shape theorem for the percolation region, saying that the
normalized region $B(t)/t$ converges to some limit shape $\mathcal{B}$. This
paper introduces a general FPP model defined on crystal lattices, and shows the
monotonicity of the limit shapes under covering maps, thereby providing insight
into the limit shape of the cubic FPP model.

Probability