## Electronic Journal of Probability

### Random surface growth and Karlin-McGregor polynomials

Theodoros Assiotis

#### Abstract

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 Borodin and Olshanski in [10] and the ones on the BC-type graph recently studied by Cuenca in [17]. Moreover, we introduce a general inhomogeneous random growth process with a wall that includes as special cases the ones considered by Borodin and Kuan [8] and Cerenzia [15], that are related to the representation theory of classical groups and also the Jacobi growth process more recently studied by Cerenzia and Kuan [16]. 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 Borodin and Olshanski in [11], and that depend on the inhomogeneities.

#### Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 106, 81 pp.

Dates
Accepted: 11 October 2018
First available in Project Euclid: 23 October 2018

https://projecteuclid.org/euclid.ejp/1540260052

Digital Object Identifier
doi:10.1214/18-EJP236

Mathematical Reviews number (MathSciNet)
MR3870449

Zentralblatt MATH identifier
06970411

Subjects
Primary: 60G

#### Citation

Assiotis, Theodoros. Random surface growth and Karlin-McGregor polynomials. Electron. J. Probab. 23 (2018), paper no. 106, 81 pp. doi:10.1214/18-EJP236. https://projecteuclid.org/euclid.ejp/1540260052

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