Sharp analysis on the joint distribution of the number of descents and inverse descents in a random permutation

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.