Half-Space Macdonald Processes - Sorbonne Université
Journal Articles Forum of Mathematics, Pi Year : 2020

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. Nonrigorous saddle-point asymptotics yield convergence of the directed polymer free energy to either the Tracy-Widom (associated to the Gaussian orthogonal or symplectic ensemble) or the Gaussian distribution depending on the average size of weights on the boundary.
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Dates and versions

hal-02796984 , version 1 (05-06-2020)

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Guillaume Barraquand, Alexei Borodin, Ivan Corwin. Half-Space Macdonald Processes. Forum of Mathematics, Pi, 2020, 8, pp.e11. ⟨10.1017/fmp.2020.3⟩. ⟨hal-02796984⟩
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