  We describe the critical behavior of weak multiplex percolation, a
generalization of percolation to multiplex or interdependent networks. A node
can determine its active or inactive status simply by referencing neighboring
nodes. This is not the case for the more commonly studied generalization of
percolation to multiplex networks, the mutually connected clusters, which
requires an interconnecting path within each layer between any two vertices in
the giant mutually connected component. We study the emergence of a giant
connected component of active nodes under the weak percolation rule, finding
several non-typical phenomena. In two layers, the giant component emerges with
a continuos phase transition, but with quadratic growth above the critical
threshold. In three or more layers, a discontinuous hybrid transition occurs,
similar to that found in the giant mutually connected component. In networks
with asymptotically powerlaw degree distributions, defined by the decay
exponent $\gamma$, the discontinuity vanishes but at $\gamma=1.5$ in three
layers, more generally at $\gamma = 1+ 1/(M-1)$ in $M$ layers.
