Probability Theory and Related Fields

, Volume 172, Issue 1–2, pp 323–385 | Cite as

Inhomogeneous exponential jump model

  • Alexei Borodin
  • Leonid Petrov


We introduce and study the inhomogeneous exponential jump model—an 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.

Mathematics Subject Classification

82C22 82B23 



The authors are grateful to discussions with Guillaume Barraquand, Ivan Corwin, Michael Blank, Tomohiro Sasamoto, Herbert Spohn, Kazumasa Takeuchi, and Jon Warren. The research was carried out in part during the authors’ participation in the Kavli Institute for Theoretical Physics program “New approaches to non-equilibrium and random systems: KPZ integrability, universality, applications and experiments”, and consequently was partially supported by the National Science Foundation PHY11-25915. A. B. is supported by the National Science Foundation grant DMS-1607901 and by Fellowships of the Radcliffe Institute for Advanced Study and the Simons Foundation.


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Institute for Information Transmission ProblemsMoscowRussia
  3. 3.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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