Half-space Macdonald processes

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.

arXiv commentv2: minor edits. 106 pages, 17 figures

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Cited by in corpus (10)

• The lower tail of the half-space KPZ equation
• The KPZ equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall
• Half-space stationary Kardar-Parisi-Zhang equation beyond the Brownian case
• Boundary current fluctuations for the half space ASEP and six vertex model
• Random walk on nonnegative integers in beta distributed random environment
• The geometric Burge correspondence and the partition function of polymer replicas
• A stationary model of non-intersecting directed polymers
• Asymptotic fluctuations of geometric q-TASEP, geometric q-PushTASEP and q-PushASEP
• Stationary measure for six-vertex model on a strip
• The second class particle in the half-line open TASEP