  We consider Bernoulli hyper-edge percolation on $\mathbb{Z}^d$. This model is
a generalization of Bernoulli bond percolation. An edge connects exactly two
vertices and a hyper-edge connects more than two vertices. As in the classical
Bernoulli bond percolation, we open hyper-edges independently in a homogeneous
manner with certain probabilities parameterized by a parameter $u\in[0,1]$. We
discuss conditions for non-trivial phase transitions when $u$ varies. We
discuss the conditions for the uniqueness of the infinite cluster. Also, we
provide conditions under which the Grimmett-Marstrand type theorem holds in the
supercritical regime.
