Anisotropic Young Diagrams and Infinite-Dimensional Diffusion Processes with the Jack Parameter

We construct a family of Markov processes with continuous sample trajectories on an infinite-dimensional space, the Thoma simplex. The family depends on three continuous parameters, one of which, the Jack parameter, is similar to the beta parameter in random matrix theory. The processes arise in a scaling limit transition from certain finite Markov chains, the so called up-down chains on the Young graph with the Jack edge multiplicities. Each of the limit Markov processes is ergodic and its stationary distribution is a symmetrizing measure. The infinitesimal generators of the processes are explicitly computed; viewed as self-adjoint operators in the L² spaces over the symmetrizing measures, the generators have a purely discrete spectrum which is explicitly described. For the special value 1 of the Jack parameter, the limit Markov processes coincide with those of the recent work by Borodin and the author (Probability Theory and Related Fields 144 (2009), 281–318). In the limit, as the Jack parameter goes to 0, our family of processes degenerates to the one-parameter family of diffusions on the Kingman simplex studied long ago by Ethier and Kurtz in connection with some models of population genetics. The techniques of the article are essentially algebraic. The main computations are performed in the algebra of shifted symmetric functions with the Jack parameter and rely on the concept of anisotropic Young diagrams due to Kerov.