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HALF-SPACE MACDONALD PROCESSES

Part of: Probability theory on algebraic and topological structures Special processes Equilibrium statistical mechanics Applications to specific types of physical systems Miscellaneous topics - Partial differential equations Algebraic combinatorics

Published online by Cambridge University Press:  26 May 2020

GUILLAUME BARRAQUAND ,
ALEXEI BORODIN  and
IVAN CORWIN
GUILLAUME BARRAQUAND
Affiliation:
Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, Paris, France; barraquand@math.cnrs.fr
ALEXEI BORODIN
Affiliation:
Department of Mathematics, MIT, Cambridge, USA Institute for Information Transmission Problems, Moscow, Russia; borodin@math.mit.edu
IVAN CORWIN
Affiliation:
Columbia University, Department of Mathematics, 2990 Broadway, New York, NY 10027, USA; ivan.corwin@gmail.com

Abstract

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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.

MSC classification

Primary: 60B99: None of the above, but in this section
Secondary: 05E05: Symmetric functions and generalizations 82B23: Exactly solvable models; Bethe ansatz 60B20: Random matrices (probabilistic aspects; for algebraic aspects see ) 82D60: Polymers 60K35: Interacting random processes; statistical mechanics type models; percolation theory 35R60: Partial differential equations with randomness, stochastic partial differential equations
Type
Probability
Information
Forum of Mathematics, Pi , Volume 8 , 2020 , e11
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

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