[Paper Review] Infinite wedge and measures on partitions

Andreĭ Okounkov|arXiv (Cornell University)|Jul 20, 1999

Advanced Combinatorial Mathematics25 citations

TL;DR

This paper establishes exact formulas for correlation functions of the Schur measure on integer partitions using integrable systems techniques, showing they are tau-functions of the Toda lattice hierarchy. It provides a new proof of Bloch-Okounkov's n-point function formula and reveals the local structure of typical partitions through this framework.

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.

Motivation & Objective

• To derive exact formulas for correlation functions of the Schur measure on integer partitions.
• To demonstrate that these correlation functions are tau-functions of the Toda lattice hierarchy.
• To provide a new proof of the Bloch-Okounkov formula for n-point functions under the uniform measure on partitions.
• To analyze the local structure of a typical partition using the developed framework.

Proposed method

• Employing techniques from integrable systems, particularly those related to the Toda lattice hierarchy.
• Defining the Schur measure as a generalization of the Plancherel measure on integer partitions.
• Deriving correlation functions using generating functions and vertex operator formalism.
• Establishing a correspondence between correlation functions and tau-functions of the Toda hierarchy.
• Applying the Boson-Fermion correspondence to relate partition functions to fermionic Fock space constructions.
• Using the infinite wedge representation to analyze the structure of the uniform measure and its n-point functions.

Experimental results

Research questions

• RQ1 How can correlation functions of the Schur measure on partitions be explicitly computed using integrable systems methods?
• RQ2 Are these correlation functions related to known hierarchies of soliton equations, such as the Toda lattice hierarchy?
• RQ3 Can the Bloch-Okounkov formula for n-point functions under the uniform measure on partitions be rederived using a new approach?
• RQ4 What is the local structure of a typical partition as revealed by the Schur measure and its correlation functions?

Key findings

• The correlation functions of the Schur measure are shown to be tau-functions of the Toda lattice hierarchy, establishing a deep link between random partitions and integrable systems.
• A new proof is provided for the Bloch-Okounkov formula for n-point functions of the uniform measure on partitions, using the framework of the infinite wedge and vertex operators.
• The Schur measure generalizes the Plancherel measure and allows for exact computation of correlation functions via integrable system techniques.
• The local structure of a typical partition is characterized through the asymptotic behavior of correlation functions, revealing universal features in the bulk.