  We study the cascading failures in a system composed of two interdependent
square lattice networks A and B placed on the same Cartesian plane, where each
node in network A depends on a node in network B randomly chosen within a
certain distance $r$ from the corresponding node in network A and vice versa.
Our results suggest that percolation for small $r$ below $r_{\rm max}\approx 8$
(lattice units) is a second-order transition, and for larger $r$ is a
first-order transition. For $r<r_{\rm max}$, the critical threshold increases
linearly with $r$ from 0.593 at $r=0$ and reaches a maximum, 0.738 for
$r=r_{\rm max}$ and then gradually decreases to 0.683 for $r=\infty$. Our
analytical considerations are in good agreement with simulations. Our study
suggests that interdependent infrastructures embedded in Euclidean space become
most vulnerable when the distance between interdependent nodes is in the
intermediate range, which is much smaller than the size of the system.
