It was recently discovered by Baik, Deift and Johansson [4] that the asymptotic distribution of the length of the longest increasing subsequence in a permutation chosen uniformly at random from Sn, properly centred and normalised, is the same as the asymptotic distribution of the largest eigenvalue of an n × n GUE random matrix, properly centred and normalised, as n → ∞. This distribution had e...

To each partition λ, we introduce a measure Sλ(x; t)sλ(y)/Zt where sλ is the Schur function and Sλ(x; t) is a generalization of the Schur function defined in [M] and Zt is a normalization constant. This measure, which we call the t-Schur measure, is a generalization of the Schur measure [O] and the shifted Schur measure studied by Tracy and Widom [TW3]. We prove that by a certain specialization...

We determined the probability distribution of the combined output power from 25 coupled fiber lasers and show that it agrees well with the Tracy-Widom and Majumdar-Vergassola distributions of the largest eigenvalue of Wishart random matrices with no fitting parameters. This was achieved with 500,000 measurements of the combined output power from the fiber lasers, that continuously changes with ...

1 For a description of this process and an explicit expression of the elements of T, see A. Wintner, Spektraltheorie der unendlichen matrizen, 1929, pp. 32-61. 2 These PROCEEDINGS, 17, 676-678 (1931). 8 This fact was pointed out to me by Dr. John Williamson of this department. This particular form of statement is convenient in a solution of the problem of determining the region of the complex p...

Example : (Local maxima) Let Z = (Z1, ..., Zn) be i.i.d. U(0, 1) random variables and let Sn be the number of local maxima in Z. Then Sn/ √ V ar(Sn)→d N (0, 1) as n→∞. CLT is so strong due to its universality and a natural question is whether there are other laws that own the same property. Tracy-Widom distribution and its variants form such a class and they appear in several statistics of rand...