Half-space Macdonald processes

TL;DR

This paper develops the theory of half-space Macdonald processes, providing new formulas and connections to stochastic models like the log-gamma polymer and ASEP, advancing understanding of half-space integrable probability models.

Contribution

It introduces the structural theory of half-space Macdonald processes, computes moments and transforms, and relates them to key stochastic models, extending integrability results to half-space settings.

Findings

Derived explicit formulas for Laplace transforms of observables.

Connected half-space Macdonald processes to the log-gamma polymer and ASEP.

Predicted Tracy-Widom and Gaussian fluctuations via saddle point analysis.

Abstract

Macdonald processes are measures on sequences of integer partitions built using the Cauchy summation identity for Macdonald symmetric functions. These measures are a useful tool to uncover the integrability of many probabilistic systems, including the Kardar-Parisi-Zhang (KPZ) equation and a number of other models in its universality class. In this paper we develop the structural theory behind half-space variants of these models and the corresponding half-space Macdonald processes. These processes are built using a Littlewood summation identity instead of the Cauchy identity, and their analysis is considerably harder than their full-space counterparts. We compute moments and Laplace transforms of observables for general half-space Macdonald measures. Introducing new dynamics preserving this class of measures, we relate them to various stochastic processes, in particular the log-gamma polymer in a half-quadrant (they are also related to the stochastic six-vertex model in a half-quadrant and the half-space ASEP). For the polymer model, we provide explicit integral formulas for the Laplace transform of the partition function. Non-rigorous saddle point asymptotics yield convergence of the directed polymer free energy to either the Tracy-Widom GOE, GSE or the Gaussian distribution depending on the average size of weights on the boundary.