We also look at square permutations from a local point of view. We consider the local topology for permutations introduced by the first author, that is the analogue of the celebrated Benjamini-Schramm convergence for graphs, and we look at the neighborhood of a random element of the permutation. We prove that uniform square permutations locally converge (in the quenched sense) to a random limiting object, described in terms of the following construction: