Arvind Ayyer: Hook-lengths of random cells in random partitions

Abstract

For an integer \(t \geq 2\), the \(t\)-core of a partition \(\lambda\) is another partition obtained by removing as many rim-hooks of size \(t\) as possible from the Young diagram of \(\lambda\). For an integer \(n\), we consider the size of the \(t\)-core of a uniformly random partition of \(n\). We determine the full distribution of this random variable as n tends to infinity. In particular, we prove that the expectation grows like \(\sqrt{n}\). We use this result to show that the probability that \(t\) divides the hook length of a uniformly random cell in a uniformly random partition of \(n\) approaches \(1/t\) as n tends to infinity. This is joint work with Shubham Sinha (UCSD).