The minimum Manhattan distance and minimum jump of permutations

The minimum jump mj(π) of π, defined by mj(π)=min1≤i≤n−1⁡|π(i+1)−π(i)|, is another natural measure in this context. The paper computes the asymptotic moments of mj(π), and the asymptotic probability that mj(π)≥d+1 for any constant d.