A Note on Convergence of Moments for Random Young Tableaux

In recent work of Baik, Deift and Rains convergence of moments was established for the limiting joint distribution of the lengths of the first k rows in random Young tableaux. The main difficulty was obtaining a good estimate for the “tail” of the distribution and this was accomplished through a highly nontrival Riemann-Hilbert analysis. Here we give a simpler derivation. A conjecture is stated which, if true, could allow the same method to be applied to certain random growth models. Since the paper of Baik, Deift and Johansson [1] in which the authors determined the limiting distribution for the length of the longest increasing subsequence in a random permutation, or equivalently the length λ1 of the first row in a random Young tableau, there have been a variety of extensions and generalizations. In [2] the same authors determined the limiting distribution of the length λ2 of the second row and Borodin, Okounkov and Olshansky in [4] and Johansson in [7] determined the limit of the joint distribution of λ1, · · · , λk for any k. The result was that for fixed x1, · · · , xk the limit lim N→∞ Pr ( λi ≤ 2 √ N + xi N , i = 1, · · · , k ) exists and equals the limiting joint distribution of the largest k eigenvalues in the Gaussian unitary ensemble of random matrices. (Here N was the number of boxes in the Young tableaux, which were given the Plancherel measure.) In the end it turned out to be the question of the asymptotics of certain Fredholm determinants, and their derivatives, associated with the operator Sn (we use the notation of [2]) acting on l(n, n+ 1, · · ·) whose matrix entries are given by Sn(j, k) = ∞