Stochastic dynamics related to Plancherel measure on partitions

Consider the standard Poisson process in the first quadrant of the Euclidean plane, and for any point (u,v) of this quadrant take the Young diagram obtained by applying the Robinson-Schensted correspondence to the intersection of the Poisson point configuration with the rectangle with vertices (0,0), (u,0), (u,v), (0,v). It is known that the distribution of the random Young diagram thus obtained is the poissonized Plancherel measure with parameter uv. We show that for (u,v) moving along any southeast-directed curve in the quadrant, these Young diagrams form a Markov chain with continuous time. We also describe these chains in terms of jump rates. Our main result is the computation of the dynamical correlation functions of such Markov chains and their bulk and edge scaling limits.