Infinite wedge and random partitions

TL;DR

This paper applies integrable systems techniques to derive exact correlation functions for random partitions under the Schur measure, revealing their connection to the Toda lattice hierarchy and providing new insights into their local structure.

Contribution

It introduces a simple formula for correlation functions of the Schur measure and demonstrates their tau-function property for the Toda lattice hierarchy.

Findings

Correlation functions of the Schur measure are tau-functions for the Toda hierarchy.

A new proof for n-point functions of the uniform measure on partitions.

Insights into the local structure of typical partitions.

Abstract

Using techniques from integrable systems, we obtain a number of exact results for random partitions. In particular, we prove a simple formula for correlation functions of what we call the Schur measure on partitions (which is a far reaching generalization of the Plancherel measure, see math.CO/9905032) and also show that these correlations functions are tau-functions for the Toda lattice hierarchy. Also we give a new proof of the formula due to Bloch and the author, see alg-geom/9712009, for the so called n-point functions of the uniform measure on partitions and comment on the local structure of a typical partition.