| We consider the limiting distribution of the number of boxes in the first and the second rows of a random Young diagram under Plancherel measure. In the random permutation context, the first row corresponds to the length of the longest increasing subsequence of a random permutation, which has been of interest since the work of Ulam and Hammersley. The resulting distributions we obtain are the same limiting distributions as the largest and the second largest eigenvalues of a random matrix taken from the Gaussian unitary ensemble, the so-called Tracy-Widom distributions. We also obtain the convergence of all moments.; The proof starts with a Toeplitz determinant expression for the Poisson generating function of the distribution given by Gessel. By general theory, this determinant can be re-expressed as a product of the leading coefficients of certain orthogonal polynomials, and following Fokas, Its and Kitaev, we consider a Riemann-Hilbert problem for the orthogonal polynomials. The asymptotics of the orthogonal polynomials can be obtained by the Deift-Zhou method for Riemann-Hilbert problems. As in earlier work of Deift, Kriecherbauer, McLaughlin, Venakides and Zhou, the associated equilibrium measure plays a critical role in the analysis.; Using similar Riemann-Hilbert method, we also obtain certain universality results for the circular ensemble in random matrix theory. |
