Random surface growth and Karlin-McGregor polynomials. (English) Zbl 1408.60091

Summary: We consider consistent dynamics for non-intersecting birth and death chains, originating from dualities of stochastic coalescing flows and one dimensional orthogonal polynomials. As corollaries, we obtain unified and simple probabilistic proofs of certain key intertwining relations between multivariate Markov chains on the levels of some branching graphs. Special cases include the dynamics on the Gelfand-Tsetlin graph considered in the seminal work of A. Borodin and G. Olshanski [J. Funct. Anal. 263, No. 1, 248–303 (2012; Zbl 1260.60149)] and the ones on the BC-type graph recently studied by C. Cuenca in [Ann. Inst. Henri Poincaré, Probab. Stat. 54, No. 3, 1359–1407 (2018; Zbl 1401.60141)]. Moreover, we introduce a general inhomogeneous random growth process with a wall that includes as special cases the ones considered by A. Borodin and J. Kuan [Commun. Pure Appl. Math. 63, No. 7, 831–894 (2010; Zbl 1193.82013)] and M. Cerenzia [“A path property of Dyson gaps, Plancherel measures for \(\mathrm{Sp}(\infty)\)”, Preprint, arXiv:1506.08742], and random surface growth], that are related to the representation theory of classical groups and also the Jacobi growth process more recently studied by M. Cerenzia and J. Kuan [“Hard-edge asymptotics of the Jacobi growth process”, Preprint, arXiv:arXiv:1608.06384]. Its most important feature is that, this process retains the determinantal structure of the ones studied previously and for the fully packed initial condition we are able to calculate its correlation kernel explicitly in terms of a contour integral involving orthogonal polynomials. At a certain scaling limit, at a finite distance from the wall, one obtains for a single level discrete determinantal ensembles associated to continuous orthogonal polynomials, that were recently introduced by A. Borodin and G. Olshanski [Commun. Math. Phys. 353, No. 2, 853–903 (2017; Zbl 1369.60031)], and that depend on the inhomogeneities.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60J27 Continuous-time Markov processes on discrete state spaces
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
Full Text: DOI arXiv Euclid


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