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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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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DOI: https://doi.org/10.1007/s00440-012-0417-4
Keywords
- Non-intersecting paths
- Infinite-dimensional Markov process
- Determinantal point process
- Gelfand–Tsetlin scheme