  Chatteerjee and Diaconis have recently shown the asymptotic normality for the
joint distribution of the number of descents and inverse descents in a random
permutation. A noteworthy point of their results is that the asymptotic
variance of the normal distribution is diagonal, which means that the number of
descents and inverse descents are asymptotically uncorrelated.The goal of this
paper is to go further in this analysis by proving a large deviation
principlefor the joint distribution. We shall show that the rate function of
the joint distributionis the sum of the rate functions of the marginal
distributions, which also means that the number of descents and inverse
descents are asymptotically independent at the large deviation level.
However,we are going to prove that they are finely dependent at the sharp large
deviation level.
