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Macdonald processes. (English) Zbl 1291.82077
The main goal of this long paper is to develop a formalism of Macdonald processes (a family of probability measures defined on Gelfand-Tsetlin patterns) and to apply it to the study of interacting particle systems and directed random polymers. The paper comprises six chapters which can be summarized as follows. (1) Introduction to Macdonald processes. (2) Definition and main properties of the Macdonald process. (3) \(q\)-Whittaker processes and its properties. (4) Whittaker processes and its properties. (5) Directed polymer in a random media. (6) Replicas and quantum many body systems. The basic result is that the \(q\)-Laplace transform of certain elements of these processes are Fredholm determinants, and loosely speaking, almost all the development of the paper is organized around this remark.

MSC:
82C22 Interacting particle systems in time-dependent statistical mechanics
82B23 Exactly solvable models; Bethe ansatz
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
33D52 Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)
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