Macdonald processes

TL;DR

This paper develops a comprehensive theory of Macdonald processes, revealing their expectations, dynamics, and connections to integrable systems, directed polymers, and the KPZ universality class, with explicit formulas and asymptotic results.

Contribution

It introduces new Markov dynamics preserving Macdonald processes and establishes their links to integrable particle systems and KPZ universality, including explicit formulas and asymptotic analysis.

Findings

Explicit evaluation of observables for Macdonald processes

Fredholm determinant formula for q-Laplace transform at t=0

Convergence to O'Connell's Whittaker process and KPZ universality

Abstract

Macdonald processes are probability measures on sequences of partitions defined in terms of nonnegative specializations of the Macdonald symmetric functions and two Macdonald parameters q,t in [0,1). We prove several results about these processes, which include the following. (1) We explicitly evaluate expectations of a rich family of observables for these processes. (2) In the case t=0, we find a Fredholm determinant formula for a q-Laplace transform of the distribution of the last part of the Macdonald-random partition. (3) We introduce Markov dynamics that preserve the class of Macdonald processes and lead to new "integrable" 2d and 1d interacting particle systems. (4) In a large time limit transition, and as q goes to 1, the particles of these systems crystallize on a lattice, and fluctuations around the lattice converge to O'Connell's Whittaker process that describe semi-discrete Brownian directed polymers. (5) This yields a Fredholm determinant for the Laplace transform of the polymer partition function, and taking its asymptotics we prove KPZ universality for the polymer (free energy fluctuation exponent 1/3 and Tracy-Widom GUE limit law). (6) Under intermediate disorder scaling, we recover the Laplace transform of the solution of the KPZ equation with narrow wedge initial data. (7) We provide contour integral formulas for a wide array of polymer moments. (8) This results in a new ansatz for solving quantum many body systems such as the delta Bose gas.