On the distribution of the length of the second row of a Young diagram under Plancherel measure
نویسنده
چکیده
We investigate the probability distribution of the length of the second row of a Young diagram of size N equipped with Plancherel measure. We obtain an expression for the generating function of the distribution in terms of a derivative of an associated Fredholm determinant, which can then be used to show that as N → ∞ the distribution converges to the Tracy-Widom distribution [TW] for the second largest eigenvalue of a random GUE matrix. This paper is a sequel to [BDJ], where we showed that as N → ∞ the distribution of the length of the first row of a Young diagram, or equivalently, the length of the longest increasing subsequence of a random permutation, converges to the Tracy-Widom distribution [TW] for the largest eigenvalue of a random GUE matrix.
منابع مشابه
ASYMPTOTICS OF q-PLANCHEREL MEASURES
In this paper, we are interested in the asymptotic size of the rows and columns of a random Young diagram under a natural deformation of the Plancherel measure coming from Hecke algebras. The first lines of such diagrams are typically of order n, so it does not fit in the context of the work of P. Biane and P. Śniady. Using the theory of polynomial functions on Young diagrams of Kerov and Olsha...
متن کاملAsymptotics of Plancherel Measures for Symmetric Groups
1.1. Plancherel measures. Given a finite group G, by the corresponding Plancherel measure we mean the probability measure on the set G∧ of irreducible representations of G which assigns to a representation π ∈ G∧ the weight (dim π)/|G|. For the symmetric group S(n), the set S(n)∧ is the set of partitions λ of the number n, which we shall identify with Young diagrams with n squares throughout th...
متن کاملStochastic Dynamics Related to Plancherel Measure on Partitions
Consider the standard Poisson process in the first quadrant of the Euclidean plane, and for any point (u, v) of this quadrant take the Young diagram obtained by applying the Robinson–Schensted correspondence to the intersection of the Poisson point configuration with the rectangle with vertices (0, 0), (u, 0), (u, v), (0, v). It is known that the distribution of the random Young diagram thus ob...
متن کاملA 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...
متن کاملOn Convergence of Moments for Random Young Tableaux and a Random Growth Model
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. The same method is used ...
متن کامل