Electronic Journal of Probability

Random surface growth and Karlin-McGregor polynomials

Theodoros Assiotis

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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

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

Received: 12 July 2018
Accepted: 11 October 2018
First available in Project Euclid: 23 October 2018

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Primary: 60G

integrable probability random surface growth determinantal point processes

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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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  • [1] N. I. Akhiezer, The classical moment problem and some related questions in analysis, Fizmat Moscow (1961), English Translation: Oliver and Boyed Ltd, Edinburgh London (1965).
  • [2] T. Assiotis, N. O’Connell, J. Warren, Interlacing Diffusions, Available from http://arxiv.org/abs/ 1607.07182, (2016).
  • [3] R. Basu, V. Sidoravicius, A. Sly, Last Passage Percolation with a Defect Line and the Solution of the Slow Bond Problem, Available from https://arxiv.org/abs/1408.3464, (2014).
  • [4] A. Borodin, I. Corwin, Macdonald processes, Probability Theory and Related Fields, Vol. 158, 225–400, (2014).
  • [5] A. Borodin, I. Corwin, T. Sasamoto From duality to determinants in q-TASEP and ASEP, Annals of Probability, Vol. 42, No. 6, 2314–2382, (2014).
  • [6] A. Borodin, P. Ferrari, Large time asymptotics of growth models on space-like paths I:PushASEP, Electronic Journal of Probability, Vol. 13, 1380–1418, (2008).
  • [7] A. Borodin, P. Ferrari, Anisotropic Growth of Random Surfaces in 2 + 1 Dimensions, Communications in Mathematical Physics, Vol. 325, 603–684, (2014).
  • [8] A. Borodin, J. Kuan, Random surface growth with a wall and Plancherel measures for $O(\infty )$, Communications on Pure and Applied Mathematics, Vol. 63, 831–894, (2010).
  • [9] A. Borodin, G. Olshanski, The boundary of the Gelfand-Tsetlin graph: A new approach, Advances in Mathematics, 230, 1738–1779, (2012).
  • [10] A. Borodin, G. Olshanski, Markov processes on the path space of the Gelfand-Tsetlin graph and on its boundary, Journal of Functional Analysis, Vol. 263, 248–303, (2012).
  • [11] A. Borodin, G. Olshanski, The ASEP and determinantal point processes, Communications in Mathematical Physics, Vol. 353, Issue 2, 853–903 (2017).
  • [12] A. Borodin, L. Petrov, Nearest neighbor Markov dynamics on Macdonald processes, Advances in Mathematics, Vol. 300, 71–155, (2016).
  • [13] A. Borodin, L. Petrov, Higher spin six vertex model and symmetric rational functions, to appear in Selecta Mathematica, 1–124, (2016).
  • [14] A. Borodin, E. Rains, Eynard-Mehta Theorem, Schur Process, and their Pfaffian Analogs, Journal of Statistical Physics, Vol. 121, Issue 3–4, 291–317 (2005).
  • [15] M. Cerenzia, A path property of Dyson gaps, Plancherel measures for $Sp(\infty )$, and random surface growth, Available from arXiv:1506.08742, (2015).
  • [16] M. Cerenzia, J. Kuan, Hard-edge asymptotics of the Jacobi growth process, Available from https://arxiv.org/abs/1608.06384, (2016).
  • [17] C. Cuenca, Markov Processes on the Duals to Infinite-Dimensional Classical Lie Groups, Available from http://arxiv.org/abs/1608.02281, (2016).
  • [18] C. Cuenca, BC type Z-measures and determinantal point processes, Available from http:// arxiv.org/abs/1701.07060, (2017).
  • [19] T. Cox, U. R$\ddot{o} $sler, A duality relation for entrance and exit laws for Markov processes, Stochastic Processes and Their Applications, Vol. 16, Issue 2, 141–156, (1984).
  • [20] P. Diaconis, J. A. Fill, Strong Stationary Times Via a New Form of Duality, Annals of Probability, Vol. 18, No. 4, 1483–1522, (1990).
  • [21] Y. Doumerc, PhD Thesis: Matrices aleatoires, processus stochastiques et groupes de reflexions, Available from http://perso.math.univ-toulouse.fr/ledoux/files/2013/11/PhD-thesis.pdf, (2005).
  • [22] A. Edrei On the generating functions of doubly infinite totally positive sequences, Transactions of the American Mathematical Society, Vol. 74 367–383, (1953).
  • [23] S. M. Fallat, C. R. Johnson Totally Nonnegative Matrices, Princeton University Press, (2011).
  • [24] P. Ferrari, H. Spohn, T. Weiss, Brownian motions with one-sided collisions: the stationary case, Electronic Journal of Probability, Vol 20, 1–41, (2015).
  • [25] S. Karlin, Total Positivity, Volume 1, Stanford University Press, (1968).
  • [26] S. Karlin, J. McGregor The classification of Birth and Death processes, Transactions of the American Mathematical Society, Vol. 86, No. 2,366–400, (1957).
  • [27] S. Karlin, J. McGregor The differential equations for Birth and Death processes and the Stieljes moment problem, Transactions of the American Mathematical Society, Vol. 85, 489–546, (1957).
  • [28] S. Karlin, J. McGregor Coincidence properties of birth and death processes, Pacific Journal of Mathematics, Vol. 9, No. 4, 1109–1140, (1959).
  • [29] S. Karlin, J. McGregor Determinants of orthogonal polynomials, Bulletin of the American Mathematical Society, Vol. 68, No. 3, 204–209, (1962).
  • [30] J. H. B Kemperman, An analytical approach to the differential equations of the birth and death process, Michigan Mathematical Journal, 9, 321–361, (1962).
  • [31] J. Kuan, A multi-species ASEP(q,j) and q-TAZRP with stochastic duality, to appear International Mathematics Research Notices, (2017).
  • [32] J. Kuan, An algebraic construction of duality functions for the stochastic $\mathcal{U} _q(A_n^{(1)})$ vertex models and its degenerations, Communications in Mathematical Physics, Vol. 359, Issue 1, 121–187, (2018).
  • [33] Y. Le Jan, O. Raimond. Flows, Coalescence and Noise, Annals of Probability, Vol. 32, No. 2, 1247–1315, (2004).
  • [34] T. Liggett, Continuous Time Markov Processes An Introduction, Graduate Studies in Mathematics, Volume 113, (2010).
  • [35] P. Nevai, Geza Freud, orthogonal polynomials and Christoffel functions, A case study, Journal of approximation theory, 48, 3–167, (1986).
  • [36] A. Okounkov, Multiplicities and Newton polytopes, in Kirillov’s seminar on Representation theory, Editor G. I. Olshanski, American Mathematical Society Translations, Series 2, Volume 181, (1998).
  • [37] A. Okounkov, G. Olshanski, Asymptotics for Jack polynomials as the number of variables goes to infinity, International Mathematics Research Notices, No. 13, 641–682, (1998).
  • [38] A. Okounkov, G. Olshanski, Limits of BC-type orthogonal polynomials as the number of variables goes to infinity, Jack, Hall-Littlewood and Macdonald Polynomials (E. B. Kuznetsov and S. Sahi editors). American Mathematical Society, Contemporary Mathematics vol. 417, (2006).
  • [39] G. Olshanski, The problem of harmonic analysis on the infinite dimensional unitary group, Journal of Functional Analysis, Vol. 205, 464–524, (2003).
  • [40] W. Pruitt, Bilateral Birth and Death Processes, Transactions of the American Mathematical Society, Vol. 107, No. 3, 508–525, (1963).
  • [41] L. C. G Rogers, J. Pitman, Markov Functions, Annals of Probability, Vol. 9, No. 4, 573–582, (1981).
  • [42] W. Schoutens, Orthogonal Polynomials and Stochastic Processes, Lecture Notes in Statistics, Springer, 2000.
  • [43] A. N. Sergeev, A. P. Veselov, $BC_{\infty }$ Calogero Moser operator and super Jacobi polynomials, Advances in Mathematics, Volume 222, Issue 5, 1687–1726, (2009).
  • [44] D. Siegmund, The Equivalence of Absorbing and Reflecting Barrier Problems for Stochastically Monotone Markov Processes, Annals of Probability, Volume 4, No. 6, 914–924, (1976).
  • [45] E. Van doorn, Stochastic monotonicity of birth and death chains, Advances in Applied Probability, Volume 12, No. 1, 59–80, (1980).
  • [46] E. Van doorn, On oscillation properties and interval of orthogonality of orthogonal polynomials, SIAM Journal of Mathematical Analysis, Volume 15, No. 5, 1031–1042, (1984).
  • [47] A. M. Vershik, S. V. Kerov Characters and factor representations of the infinite unitary group, Dokl. Akad. Nauk SSSR 267(2) (1982), 272–276 (in Russian); English Translation: Soviet Math. Dokl. 26, 570–574, (1982).
  • [48] D. Voiculescu, Representations factorielles de type $II_{1}$ de $U(\infty )$, Journal de Mathematiques Pures et Appliquees, 55, 1–20, (1976).
  • [49] Z. Wang, X. Yang Birth and Death Processes and Markov Chains, Springer, (1992).
  • [50] J. Warren, Dyson’s Brownian motions,intertwining and interlacing, Electronic Journal of Probability, Vol.12, 573–590, (2007).
  • [51] J. Warren, P. Windridge Some Examples of Dynamics for Gelfand-Tsetlin Patterns, Electronic Journal of Probability, Vol.14, 1745–1769, (2009).