Rectangular Young tableaux and the Jacobi ensemble

Abstract

International audience It has been shown by Pittel and Romik that the random surface associated with a large rectangular Youngtableau converges to a deterministic limit. We study the fluctuations from this limit along the edges of the rectangle.We show that in the corner, these fluctuations are gaussian whereas, away from the corner and when the rectangle isa square, the fluctuations are given by the Tracy-Widom distribution. Our method is based on a connection with theJacobi ensemble.

Periodic Pólya urns, the density method and asymptotics of Young tableaux

“…Local limit laws see e.g. [15,66] & Gaussian for rectangular shapes [64], see also [49] for some special cases see also [38] Fig 7 . Known and conjectured limit laws of random Young tableaux. Would it one day lead to a nice notion of "continuous Young tableau"?…”

“…it has shape λ i 1 1 ), the associated tree has shape (1, (λ 1 − 1)(i 1 − 1); i 1 , λ 1 − 1). In that case, the law of E T (v m ) is easy to compute and we get an alternative proof of the following formula, first established in [64]:…”

“…Many results on their asymptotic shape have been collected, but very few results are known on their asymptotic content when the shape is fixed (see e.g. the works by Pittel and Romik, Angel et al, Marchal [4,64,74,82], who have studied the distribution of the values of the cells in random rectangular or staircase Young tableaux, while the case of Young tableaux with a more general shape seems to be very intricate). It is therefore pleasant that our work on periodic Pólya urns allows us to get advances on the case of a triangular shape, with any rational slope.…”

“…proven in [52] for random tilings, see also [14,73,74] Generalized gamma product distribution local limit law proven in this article, Airy ensemble? proven in [64] for square shapes (includes Tracy-Widom distribution),…”

Periodic Pólya Urns and an Application to Young Tableaux

“…Many results on their asymptotic shape have been collected, but very few results are known on their asymptotic content when the shape is fixed (see e.g. the works by Pittel and Romik, Angel et al, Marchal [1,24,26,29], who have studied the distribution of the values of the cells in random rectangular or staircase Young tableaux, while the case of Young tableaux with a more general shape seems to be very intricate). It is therefore pleasant that our work on periodic Pólya urns allows us to get advances on the case of a triangular shape, with any slope.…”