The limiting distribution of the hook length of a randomly chosen cell in a random Young diagram

Abstract

Let $p(n)$ be the number of all integer partitions of the positive integer $n$ and let $\lambda$ be a partition, selected uniformly at random from among all such $p(n)$ partitions. It is known that each partition $\lambda$ has a unique graphical representation, composed by $n$ non-overlapping cells in the plane called Young diagram. As a second step of our sampling experiment, we select a cell $c$ uniformly at random from among all $n$ cells of the Young diagram of the partition $\lambda$. For large $n$, we study the asymptotic behavior of the hook length $Z_n=Z_n(\lambda,c)$ of the cell $c$ of a random partituion $\lambda$. This two-step sampling procedure suggests a product probability measure, which assigns the probability $1/np(n)$ to each pair $(\lambda,c)$. With respect to this probability measure, we show that the random variable $\pi Z_n/\sqrt{6n}$ converges weakly, as $n\to\infty$, to a random variable whose probability density function equals $6y/\pi^2 (e^y-1)$ if $0<y<\infty$, and zero elsewhere.