On Increasing Subsequences of I.i.d. Samples

We study the uctuations, in the large deviations regime, of the longest increasing subsequence of a random i.i.d. sample on the unit square. In particular, our results yield the precise upper and lower exponential tails for the length of the longest increasing subsequence of a random permutation. i=1 denote a sequence of i.i.d. random variables with marginal law on the unit square Q = 0; 1] 2. Throughout, we make the assumption that possesses a strictly positive density p 2 C 1 (Q) with respect to the Lebesgue measure on Q. Deene next`max (n) to be the length of the longest increasing subsequence in the sample fZ i g n i=1. Note that we do not require that i j < i j+1. In case that = , ` max (n) possesses the same law as the length of the longest increasing subsequence of a random permutation, denoted hereafter by L max (n). Building on the fact that lim n!1 L max (n) p n = 2 in probability; cf. 8], we showed in 3] that (1.1) lim n!1