pr.probability: Probability that the perfect match is not nested on the N route with the given interval?

Given a trajectory graph $ P_n $ ($ n $ even). We add new set of $ n / 2 $ borders $ C $ between non-adjacent path nodes such that establish $ C $ It forms a perfect combination. The output graph is the union of the route. $ P_n $ and perfect match $ C $. Each edge $ e $ connects two non-adjacent nodes $ V_i $ Y $ V_j $ in the path $ P_n $. The lapse of an edge $ e (i, j) $ in perfect match it's the same $ abs (i-j) $. The perfect match span $ C $ is $ max_ {e in C} span (e) $ .

What is the probability that there is a perfect match without nesting when the perfect match interval is at most? $ n ^ r $ where $ r lt 1 $?

This was motivated by this post.