Let π be a permutation of . If we identify a permutation with its graph, namely the set of n dots at positions , it is natural to consider the minimum (Manhattan) distance, , between any pair of dots. The paper computes the expected value (and higher moments) of when and π is chosen uniformly, and settles a conjecture of Bevan, Homberger and Tenner (motivated by permutation patterns), showing that when d is fixed and , the probability that tends to .
The minimum jump of π, defined by , is another natural measure in this context. The paper computes the asymptotic moments of , and the asymptotic probability that for any constant d.