What is it about?

Take a chessboard. For each column, toss a coin: heads, you put a pawn somewhere in that column — tails, you don't. Do this in any way you like. You may look at all coin flips before choosing where you put your first pawn. You may also use additional randomness if you want to. Our result states that it is then possible for me to put additional pawns, but never touching yours, in such a way that the final configuration behaves exactly as if we decided to flip a coin for each square and put a pawn at each square with the result "heads". We study diverse generalisations and variations of this result.

By using the aforementioned result, we revisit a well known result of Benjamini and Schramm stating that quotienting a graph can never increase its percolation threshold. We also revisit a strict version of this monotonicity result, due to Franco Severo and myself. For this second result, we do not use these pawn-on-chessboard techniques but explorations and couplings. Contrary to the original proof, our new argument does not make any use of essential enhancements or differential inequalities.

Arxiv:2504.02427