Inhomogeneous exponential jump model

• 成果类型: Article
• 署名作者: Borodin, Alexei; Petrov, Leonid
• 署名单位: Massachusetts Institute of Technology (MIT); Kharkevich Institute for Information Transmission Problems of the RAS; University of Virginia
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0810-0
• 发表日期: 2018
• 页码: 323-385
• 关键词: yang-baxter equation; particle system; distributions; fluctuations; asymptotics; tasep; limit

摘要

We introduce and study the inhomogeneous exponential jump modelan integrable stochastic interacting particle system on the continuous half line evolving in continuous time. An important feature of the system is the presence of arbitrary spatial inhomogeneity on the half line which does not break the integrability. We completely characterize the macroscopic limit shape and asymptotic fluctuations of the height function (=integrated current) in the model. In particular, we explain how the presence of inhomogeneity may lead to macroscopic phase transitions in the limit shape such as shocks or traffic jams. Away from these singularities the asymptotic fluctuations of the height function around its macroscopic limit shape are governed by the GUE Tracy-Widom distribution. A surprising result is that while the limit shape is discontinuous at a traffic jam caused by a macroscopic slowdown in the inhomogeneity, fluctuations on both sides of such a traffic jam still have the GUE Tracy-Widom distribution (but with different non-universal normalizations). The integrability of the model comes from the fact that it is a degeneration of the inhomogeneous stochastic higher spin six vertex models studied earlier in Borodin and Petrov(Higher spin six vertex model and symmetric rational functions, doi:10.1007/s00029-016-0301-7, arXiv:1601.05770 [math.PR], 2016). Our results on fluctuations are obtained via an asymptotic analysis of Fredholm determinantal formulas arising from contour integral expressions for the q-moments in the stochastic higher spin six vertex model. We also discuss product-form translation invariant stationary distributions of the exponential jump model which lead to an alternative hydrodynamic-type heuristic derivation of the macroscopic limit shape.