Inhomogeneous exponential jump model
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 th...
| Main Authors: | , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer Berlin Heidelberg
2018
|
| Online Access: | http://hdl.handle.net/1721.1/117684 https://orcid.org/0000-0002-2913-5238 |
| _version_ | 1811085412979965952 |
|---|---|
| author | Borodin, Alexei Petrov, Leonid |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Borodin, Alexei Petrov, Leonid |
| author_sort | Borodin, Alexei |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T13:09:11Z |
| format | Article |
| id | mit-1721.1/117684 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T13:09:11Z |
| publishDate | 2018 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1176842022-09-28T12:16:33Z Inhomogeneous exponential jump model Borodin, Alexei Petrov, Leonid Massachusetts Institute of Technology. Department of Mathematics Borodin, Alexei 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. National Science Foundation (U.S.) (Grant DMS-1607901) Simons Foundation (Fellowship) Radcliffe Institute for Advanced Study (Fellowship) 2018-09-07T20:21:41Z 2018-10 2018-09-07T03:47:54Z Article http://purl.org/eprint/type/JournalArticle 0178-8051 1432-2064 http://hdl.handle.net/1721.1/117684 Borodin, Alexei, and Leonid Petrov. “Inhomogeneous Exponential Jump Model.” Probability Theory and Related Fields, vol. 172, no. 1–2, Oct. 2018, pp. 323–85. https://orcid.org/0000-0002-2913-5238 en https://doi.org/10.1007/s00440-017-0810-0 Probability Theory and Related Fields Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ Springer-Verlag GmbH Germany application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Borodin, Alexei Petrov, Leonid Inhomogeneous exponential jump model |
| title | Inhomogeneous exponential jump model |
| title_full | Inhomogeneous exponential jump model |
| title_fullStr | Inhomogeneous exponential jump model |
| title_full_unstemmed | Inhomogeneous exponential jump model |
| title_short | Inhomogeneous exponential jump model |
| title_sort | inhomogeneous exponential jump model |
| url | http://hdl.handle.net/1721.1/117684 https://orcid.org/0000-0002-2913-5238 |
| work_keys_str_mv | AT borodinalexei inhomogeneousexponentialjumpmodel AT petrovleonid inhomogeneousexponentialjumpmodel |