Random Matrices and Random Permutations

TL;DR

This paper proves a conjecture linking the behavior of partition rows under Plancherel measure to eigenvalues of Gaussian Hermitian matrices, using surface maps and moduli space intersection theory.

Contribution

It establishes a new connection between random partitions, surface topology, and algebraic geometry, proving a conjecture about asymptotic eigenvalue distributions.

Findings

Partition rows under Plancherel measure converge to eigenvalues of Gaussian matrices

Connection between surface maps and random matrix theory

Link to intersection theory on moduli spaces of curves

Abstract

We prove the conjecture of Baik, Deift, and Johansson which says that with respect to the Plancherel measure on the set of partitions of $n$, the 1st, 2nd, and so on, rows behave, suitably scaled, like the 1st, 2nd, and so on, eigenvalues of a Gaussian random Hermitian matrix as $n$ goes to infinity. Our proof is based on an interplay between maps on surfaces and ramified coverings of the sphere. We also establish a connection of this problem with intersection theory on the moduli spaces of curves.