Operators and Markov Processes on Branching Graphs

2011/11/15

Journal of Algebraic Combinatorics 38 (2013), no. 3, 663-720arXiv:1111.3399 [math.CO]

We present a unified approach to various examples of Markov dynamics on partitions studied by Borodin, Olshanski, Fulman, and the author. Our technique generalizes the Kerov’s operators first appeared in [Okounkov, arXiv:math/0002135], and also stems from the study of duality of graded graphs in [Fomin, 1994].

Our main object is a countable branching graph carrying an $\mathfrak{sl}(2,\mathbb{C})$-module of a special kind. Using this structure, we introduce distinguished probability measures on the floors of the graph, and define two related types of Markov dynamics associated with these measures. We study spectral properties of the dynamics, and our main result is the explicit description of eigenfunctions of the Markov generator of one of the processes.

For the Young graph our approach reconstructs the z-measures on partitions and the associated dynamics studied by Borodin and Olshanski [arXiv:math-ph/0409075, arXiv:0706.1034]. The generator of the dynamics of [arXiv:math-ph/0409075] is diagonal in the basis of the Meixner symmetric functions introduced recently by Olshanski [arXiv:1009.2037, arXiv:1103.5848]. We give new proofs to some of the results of these two papers. Other graphs to which our technique is applicable include the Pascal triangle, the Kingman graph (with the two-parameter Poisson-Dirichlet measures), the Schur graph and the general Young graph with Jack edge multiplicities.