Some stochastic processes in a random permutation

Abstract

For a random permutation σ of {1,…, N} an increasing success at i is the event that σ( i)= α, σ( h)= α−1 for some h< i and α≧2 and a decreasing success the event that σ( i)= β, σ( k)= β+1 for some k< i and β≦ N−1. Three other processes of events have the same distribution as these successes, the first one defined in terms of rises and descents of σ, the second one on a random circular permutation of {1,…, N+1} and the third one on a chess board model. The process of successes of both types has the Markov property with respect to the total number of preceding successes. The distribution of some statistics of these processes are given and their asymptotic behaviour as N → ∞ is studied.