Non-intersecting path constructions for TASEP with inhomogeneous rates and the KPZ fixed point
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911392535150592 |
|---|---|
| author | Bisi, Elia Liao, Yuchen Saenz, Axel Zygouras, Nikos |
| author_facet | Bisi, Elia Liao, Yuchen Saenz, Axel Zygouras, Nikos |
| contents | We consider a discrete-time TASEP, where each particle jumps according to Bernoulli random variables with particle-dependent and time-inhomogeneous parameters. We use the combinatorics of the Robinson-Schensted-Knuth correspondence and certain intertwining relations to express the transition kernel of this interacting particle system in terms of ensembles of weighted, non-intersecting lattice paths and, consequently, as a marginal of a determinantal point process. We next express the joint distribution of the particle positions as a Fredholm determinant, whose correlation kernel is given in terms of a boundary-value problem for a discrete heat equation. The solution to such a problem finally leads us to a representation of the correlation kernel in terms of random walk hitting probabilities, generalising the formulation of Matetski, Quastel and Remenik (Acta Math., 2021) to the case of both particle- and time-inhomogeneous rates. The solution to the boundary value problem in the fully inhomogeneous case appears with a finer structure than in the homogeneous case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_13580 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Non-intersecting path constructions for TASEP with inhomogeneous rates and the KPZ fixed point Bisi, Elia Liao, Yuchen Saenz, Axel Zygouras, Nikos Probability Mathematical Physics Combinatorics 05E05, 60Cxx, 82B23 We consider a discrete-time TASEP, where each particle jumps according to Bernoulli random variables with particle-dependent and time-inhomogeneous parameters. We use the combinatorics of the Robinson-Schensted-Knuth correspondence and certain intertwining relations to express the transition kernel of this interacting particle system in terms of ensembles of weighted, non-intersecting lattice paths and, consequently, as a marginal of a determinantal point process. We next express the joint distribution of the particle positions as a Fredholm determinant, whose correlation kernel is given in terms of a boundary-value problem for a discrete heat equation. The solution to such a problem finally leads us to a representation of the correlation kernel in terms of random walk hitting probabilities, generalising the formulation of Matetski, Quastel and Remenik (Acta Math., 2021) to the case of both particle- and time-inhomogeneous rates. The solution to the boundary value problem in the fully inhomogeneous case appears with a finer structure than in the homogeneous case. |
| title | Non-intersecting path constructions for TASEP with inhomogeneous rates and the KPZ fixed point |
| topic | Probability Mathematical Physics Combinatorics 05E05, 60Cxx, 82B23 |
| url | https://arxiv.org/abs/2208.13580 |