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Markov processes of infinitely many nonintersecting random walks
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  • Published: 14 March 2012

Markov processes of infinitely many nonintersecting random walks

  • Alexei Borodin1,2,3 &
  • Vadim Gorin3,4 

Probability Theory and Related Fields volume 155, pages 935–997 (2013)Cite this article

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Abstract

Consider an N-dimensional Markov chain obtained from N one-dimensional random walks by Doob h-transform with the q-Vandermonde determinant. We prove that as N becomes large, these Markov chains converge to an infinite-dimensional Feller Markov process. The dynamical correlation functions of the limit process are determinantal with an explicit correlation kernel. The key idea is to identify random point processes on \({\mathbb Z}\) with q-Gibbs measures on Gelfand–Tsetlin schemes and construct Markov processes on the latter space. Independently, we analyze the large time behavior of PushASEP with finitely many particles and particle-dependent jump rates (it arises as a marginal of our dynamics on Gelfand–Tsetlin schemes). The asymptotics is given by a product of a marginal of the GUE-minor process and geometric distributions.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Alexei Borodin

  2. Department of Mathematics, California Institute of Technology, Pasadena, CA, USA

    Alexei Borodin

  3. Dobrushin Mathematics Laboratory, Institute for Information Transmission Problems of Russian Academy of Sciences, Moscow, Russia

    Alexei Borodin & Vadim Gorin

  4. Mathematical Sciences Research Institute, Berkeley, CA, USA

    Vadim Gorin

Authors
  1. Alexei Borodin
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  2. Vadim Gorin
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Corresponding author

Correspondence to Vadim Gorin.

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Borodin, A., Gorin, V. Markov processes of infinitely many nonintersecting random walks. Probab. Theory Relat. Fields 155, 935–997 (2013). https://doi.org/10.1007/s00440-012-0417-4

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  • Received: 12 August 2011

  • Accepted: 20 February 2012

  • Published: 14 March 2012

  • Issue Date: April 2013

  • DOI: https://doi.org/10.1007/s00440-012-0417-4

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Keywords

  • Non-intersecting paths
  • Infinite-dimensional Markov process
  • Determinantal point process
  • Gelfand–Tsetlin scheme

Mathematics Subject Classification

  • 60J25
  • 60G55
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