Determinantal Transition Kernels for Some Interacting Particles on the Line

Non-colliding Markov processes are canonical examples of stochastic processes with a determinantal transition kernel, given by the Karlin-McGregor formula. A determinantal transition kernel of a different form, yet similar to the Karlin-McGregor kernel, was encountered by Schütz [15] in his study of the totally asymmetric simple exclusion process. This work has stimulated much recent research, e.g., [2, 8, 12, 14, 19, 20]. In this note we explicitly connect Schütz type formulae for particle systems and KarlinMcGregor type formulae for non-colliding processes. Our approach builds upon deep connections between particle processes and non-colliding processes (or random-matrix theory), which have been recently discovered [1, 4, 7, 11]. A combinatorial correspondence known as the Robinson-Schensted-Knuth (RSK) correspondence links these processes. This RSK correspondence gives a coupling of the non-colliding process and the particle process. Our key contributions are that this coupling implies an intertwining of their transition semigroups, and that this intertwining is enough to find the transition kernel of the particle process. We give a number of new formulae by applying this method to four variants of the RSK correspondence. Augmented with systems arising from suitable limiting procedures, the particle systems we treat in this way are known to play pivotal roles in a wide range of interesting applied problems. For instance, they appear in the context of queues in series, last-passage percolation, growth models, and fragmentation models. This note is organized as follows. In Section 2, we introduce four interacting particle systems and we present the associated transition kernels. Section 3 describes the four variants of the RSK correspondence we use in our analysis, and derives the aforementioned intertwining of the semigroups. Finally, it is the topic Section 4 to use this intertwining for finding the transition mechanism of the interacting particles.