For an infinite n-dimensional discrete grid. Each cell has probability $p$ of being on. No infinitely large connected piece? A piece is called connected if it's made up of on cells, and let C be the set of on cells in the piece, and $\forall c \in C\ \exists \ d \in C$ such that $c$ and $d$ are directly connected, that is their coordinates are the same, except for one value which differs by one.
The fact that there cannot be a connected piece of infinite size seems apparent to me. But it was debated. We agreed that the 1D case is clearly true i.e. you can't have a piece of infinite size. And the "curse of dimensionality" kinda rings a bell here since the higher the dimension, the more sparse your grid and hence even less likely to have a connected piece.
So I am seeking proof or a theorem from somewhere that can settle this once and for all.
 A: If I understand the question correctly, you're asking about site percolation on $\mathbb{Z}^d$, $d \ge 2$, and the question is whether there exists an infinite cluster.
The answer is that for $p$ close enough to 1, there will exist an infinite cluster, almost surely (i.e. with probability 1).  Indeed, there is a critical probability $p_c(d)$ such that if $p < p_c$ then almost surely there is no infinite cluster, and if $p > p_c$ then almost surely there is at least one (indeed it can be shown that almost surely there is exactly one).  This much follows from the Kolmogorov 0-1 law, since existence of an infinite cluster is a tail event.  It can be shown for $d \ge 2$ that $0 < p_c < 1$.  See for instance Grimmett's book Percolation.  Most of the book is about bond percolation, where you switch edges instead of cells, but many results can be converted from one to the other.  For this particular result, see Theorem 1.10 for bond percolation, and Theorem 1.33 for the corollary for site percolation.
Note that higher dimension makes it easier to have an infinite cluster, for any given value of $p$; for example, $\mathbb{Z}^3$ contains infinitely many copies of $\mathbb{Z}^2$, any one of which could have an infinite cluster.
There is lots of other research on percolation; again, Grimmett's book is a great place to start.
