$\mathfrak{sl}(2)$ operators and Markov processes on branching graphs

• Authors: Leonid Petrov
• Venue: Journal of Algebraic Combinatorics: An International Journal
• Year: 2013

We present a unified approach to various examples of Markov dynamics on partitions studied by Borodin, Olshanski, Fulman, and the author. Our technique generalizes Kerov's operators which first appeared in Okounkov (Random Matrix Models and Their Applications, pp. 407---420, Cambridge University Press, Cambridge, 2001), and also stems from the study of duality of graded graphs in Fomin (J. Algebr. Comb., 3(4):357---404, 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 (Probab. Theory Relat. Fields, 135(1):84---152, 2006; Probab. Theory Relat. Fields, 144(1):281---318, 2009). The generator of the jump dynamics on the Young graph corresponding to the z-measures is diagonal in the basis of the Meixner symmetric functions introduced recently by Olshanski (Zap. Nauă驴. Semin. POMI, 378:81---100, 230, 2010; Laguerre and Meixner orthogonal bases in the algebra of symmetric functions, 2011). 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.