You can always cross an $n\times n$ grid with random diagonals, from one side of the grid to the opposite side of the grid. So the probability of this crossing is $1$.
Here random diagonals means you have an $n\times n$ grid and you draw at random one diagonal in each of the 1×1 unit squares of the grid. Then you can always find a connected path using these small diagonals that goes from one side of the grid to the opposite side, up to down or left to right.
(You can prove that by contradiction that makes repeated use of the Lemma of Sperner. In these two posts, you can find a related discussion and several different proof ideas https://math.stackexchange.com/a/3677664/782412 and https://mathoverflow.net/q/112067/156936.)
I have two questions about the crossing probability
(1) Can we use the symmetry to conclude that the probability of crossing from top to down is $1/2$ ? My thinking is that this follows directly from the symmetry, and from the fact that the unrestricted (up-down or left-right) probability is $1$.
I have seen such a symmetry argument in the post of user joriki here https://math.stackexchange.com/a/3641146/782412, but I wanted to ask for confirmation that the symmetry arguments is also valid for my problem.
(2) Bond percolation critical probability threshold $p_c$? Assuming the answer to my question is $1/2$, i.e. probability of crossing the grid top-down, this reminds me of an introductory article about percolation theory here https://en.wikipedia.org/wiki/Percolation_theory, following a comment from user joriki (thank you for that!).
In particular, I am refering to bond percolation, i.e. percolating from the top side down to the bottom side. The article says that for the infinite square lattice $\mathbb Z^2$ in two dimensions, the critical probability threshold $p_c$ for bond percolation is $1/2$.
My question, is crossing the grid along random diagonals in fact EQUIVALENT to bond percolation on a square grid? In this case, my crossing problem could be related to $p_c$ for the finite case of an $n \times n$ grid? Is the finite $n\times n$ case an established and known result? I am not sure, and maybe I am misunderstanding the concept of $p_c$. I would be grateful if someone could help me clarify and answer this.