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Published April 2013 | public
Journal Article

Markov processes of infinitely many nonintersecting random walks

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 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.

Additional Information

© 2012 Springer Verlag. Received: 12 August 2011. Accepted: 20 February 2012. Published online: 14 March 2012. The authors would like to thank the anonymous referee for his valuable suggestions which improved the text. A.B. was partially supported by NSF grant DMS-1056390. V.G. was partially supported by "Dynasty" foundation, by RFBR—CNRS grant 10-01-93114, by the program "Development of the scientific potential of the higher school" and by Simons Foundation—IUM scholarship.

Additional details

Created:
August 22, 2023
Modified:
October 23, 2023