Free field approach to the Macdonald process

TL;DR

This paper introduces a free field approach to analyze the Macdonald process, revealing determinantal structures and providing new methods for computing correlation functions using algebraic and fermionic techniques.

Contribution

It develops a free field realization of the Macdonald process, enabling explicit computation of correlation functions and connecting to fermionic interpretations at the Schur limit.

Findings

Determinantal structure of correlation functions is clarified via free field realization.

Method for computing correlation functions using Heisenberg algebra and free fermions.

Proposes a generalized Macdonald measure based on Hopf algebra structures.

Abstract

The Macdonald process is a stochastic process on the collection of partitions that is a $(q,t)$-deformed generalization of the Schur process. In this paper, we approach the Macdonald process identifying the space of symmetric functions with a Fock representation of a Heisenberg algebra. By using the free field realization of operators diagonalized by the Macdonald symmetric functions, we propose a method of computing several correlation functions with respect to the Macdonald process. It is well-known that expectation value of several observables for the Macdonald process admit determinantal expression. We find that this determinantal structure is apparent in free field realization of the corresponding operators and, furthermore, it has a natural interpretation in the language of free fermions at the Schur limit. We also propose a generalized Macdonald measure motivated by recent studies on generalized Macdonald functions whose existence relies on the Hopf algebra structure of the Ding--Iohara--Miki algebra.