TITLE: Probability of a black path on a random chess board
QUESTION [8 upvotes]: Take a $2n$ by $2n$ chess board (oriented so the grid lines are vertical and horizontal). Usually there are $2n^2$ squares coloured black and $2n^2$ squares coloured white so that a black square is only adjacent to white squares. (Here, two squares are adjacent if they have a common edge.)
Suppose instead we start with a blank $2n$ by $2n$ chess board. We pick $2n^2$ squares at random and assign them black. The other half of the squares are assigned white.

What is the probability the resulting chessboard has a monotonic black path? (Here, a monotonic black path is one which starts in the South-West corner and finishes in the North-East corner, and consists entirely of black squares adjacent along their North or East edge.
What is the probability that the resulting chessboard has a black path from the South-West corner to the North-East corner?  (Here, a black path is a sequence of adjacent black squares)

REPLY [6 votes]: James quickly gave the right answer in the comments, since $p_c \approx .5927$ for site percolation on the square lattice.
These crossing questions often have elementary answers, but neither the proofs nor the applications are trivial.  For example, in critical percolation, the Russo-Seymour-Welsh theorem states that there is a uniform lower bound in the crossing probability.  i.e., there is a uniform constant $c$ such that $\mathbb P_n(\mbox{there is a black crossing}) \ge c$, independently of $n$.
For a nice proof of the RSW theorem (with illustrative pictures!), see pages 33-44 of Pierre Nolin's lecture notes.  (After deriving RSW, Pierre uses this formula to prove Kesten's theorem: $p_c = 1/2$ for bond percolation on the square lattice)
Another place to look is Section 1.3 of  Wendelin Werner's lecture notes on percolation.  Werner uses this to prove the Cardy-Smirnov formula, and then that site percolation on triangular lattice converges to $\operatorname{SLE}(6)$.
Cardy's formula is just one of the many elegant results in mathematical conformal field theory.  Define $$f(x) = \mathbb P( \mbox{crossing starting from the point $x$ on side $1$ to side $2$} )$$ for site percolation on the unit triangle with spacing $1/n$.  Cardy's formula is that $$f(x) = x.$$  (Peter Jones has described Cardy's formula as "the most difficult theorem about the identity function.")