  We study the mutual percolation of two interdependent lattice networks
ranging from two to seven dimensions, denoted as $D$. We impose that the length
of interdependent links connecting nodes in the two lattices be less than or
equal to a certain value, $r$. For each value of $D$ and $r$, we find the
mutual percolation threshold, $p_c[D,r]$ below which the system completely
collapses through a cascade of failures following an initial destruction of a
fraction $ (1-p)$ of the nodes in one of the lattices. We find that for each
dimension, $D<6$, there is a value of $r=r_I>1$ such that for $r\geq r_I$ the
cascading failures occur as a discontinuous first order transition, while for
$r<r_I$ the system undergoes a continuous second order transition, as in the
classical percolation theory. Remarkably, for $D=6$, $r_I=1$ which is the same
as in random regular (RR) graphs with the same degree (coordination number) of
nodes. We also find that in all dimensions, the interdependent lattices reach
maximal vulnerability (maximal $p_c[D,r]$) at a distance $r=r_{max}>r_I$, and
for $r>r_{max}$ the vulnerability starts to decrease as $r\to\infty$. However
the decrease becomes less significant as $D$ increases and
$p_c[D,r_{max}]-p_c[D,\infty]$ decreases exponentially with $D$. We also
investigate the dependence of $p_c[D,r]$ on the system size as well as how the
nature of the transition changes as the number of lattice sites, $N\to\infty$.
