I have nothing substantive to say, but I thought it might be helpful to
include an example of random
diagonals, which I have
drawn from an earlier MO question, "[Shortest grid-graph paths with random diagonal shortcuts][1]":
<br />&nbsp;&nbsp;&nbsp;
![Random Diagonals 50x50][2]
<br />


  [1]: https://mathoverflow.net/questions/45920/
  [2]: https://i.sstatic.net/3r34w.jpg