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Nearest neighbor Markov dynamics on Macdonald processes. (English) Zbl 1356.60161

The Macdonald process was introduced by the first author and I. Corwin in [Probab. Theory Relat. Fields 158, No. 1–2, 225–400 (2014; Zbl 1291.82077)]. The state space of the process is the set of triangular arrays \(\Lambda=\{\lambda_j^{(k)}:1\leq j\leq k\leq N\}\) (of a given depth \(N\)) of nonnegative integers subject to the following interlacing constraints: \[ \lambda_{j+1}^{(k)}\leq \lambda_j^{(k-1)}\leq \lambda_j^{(k)}. \] On the set of arrays \(\Lambda\), a probability measure is defined such that its marginals on the set of \(N\) rows of \(\Lambda\) are given by the Macdonald measure, which assigns the weight \(\mathbf{MM}(a_1,\ldots,a_k;\rho_{\tau})\) to the \(k\)-th row in \(\Lambda\). Here, \(a_1,\ldots,a_k\) are positive parameters and \(\rho_{\tau}\) plays the role of time, while the rows of \(\Lambda\) can be treated as rows of the Young diagram, corresponding to an integer partition. The Macdonald measure induces a continuous-time Markov dynamic of the corresponding process. It is known that the aforementioned probability measure on \(\Lambda\) is a Gibbs measure. Treating the considered process as a Markov jump process, the authors restrict their study to the class of nearest neighbor dynamics in which the move of any particle \(\lambda_j^{(k-1)}\) affects only the closest upper neighbors in \(\Lambda\).
The main result of the paper states that any nearest neighbor dynamic is a mixture of the fundamental nearest neighbor dynamics, which are shown to belong to three families, explicitly described in the paper. Based on this result, the authors obtain a complete classification of the continuous-time nearest neighbor “Markov dynamics” that have prescribed fixed time marginals and prescribed evolution along certain one-dimensional sections.
The authors also discover new examples of nearest neighbor dynamics.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J25 Continuous-time Markov processes on general state spaces
60J75 Jump processes (MSC2010)
82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics

Citations:

Zbl 1291.82077
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References:

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