Asymptotics of the Gelfand models of the symmetric groups

Abstract: If a partition $\lambda$ of size n is chosen randomly according to the Plancherel measure $P_n[\lambda] = (\dim \lambda)^2/n!$, then as n goes to infinity, the rescaled shape of $\lambda$ is with high probability very close to a non-random continuous curve $\Omega$ known as the Logan-Shepp-Kerov-Vershik curve. Moreover, the rescaled deviation of $\lambda$ from this limit shape can be described by an explicit generalized gaussian process. In this paper, we investigate the analoguous problem when $\lambda$ is chosen with probability proportional to $\dim \lambda$ instead of $(\dim \lambda)^2$. We shall use very general arguments due to Ivanov and Olshanski for the first and second order asymptotics (cf. arXiv:math/0304010); these arguments amount essentially to a method of moments in a noncommutative setting. The first order asymptotics of the Gelfand measures turns out to be the same as for the Plancherel measure; on the contrary, the fluctuations are different (and bigger), although they involve the same generalized gaussian process. Many of our computations relie on the enumeration of involutions and square roots in $S_n$.