A Limit Theorem for Stochastically Decaying Partitions at the Edge

TL;DR

This paper proves that stochastically decaying partitions, when scaled appropriately, converge to the Airy2 line ensemble, using a combinatorial approach that offers new insights beyond determinantal methods.

Contribution

It introduces a new combinatorial proof for the convergence of stochastically decaying partitions to the Airy2 line ensemble, expanding the analytical tools available.

Findings

Convergence of partition rows to the Airy2 line ensemble.

A combinatorial approach applicable without determinantal structure.

Highlights similarities between random partitions and random matrices.

Abstract

In this paper, we study the asymptotic behavior of the first, second, and so on rows of stochastically decaying partitions. We establish that, with appropriate scaling in time and length, the sequence of rows converges to the Airy$_2$ line ensemble. This result was first established, in a more general setting, by Borodin and Olshanski, who relied on the determinantal structure of the Poissonized correlation functions. Our argument is based on a different, combinatorial approach, developed by Okounkov. This approach may be useful in other problems in which no determinantal structure is available, and also highlights the similarity between random partitions and random matrices.