Multipoint distributions of the KPZ fixed point with compactly supported initial conditions
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2025
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| author | Liao, Yuchen Liu, Zhipeng |
| author_facet | Liao, Yuchen Liu, Zhipeng |
| contents | The KPZ fixed point is a universal limiting space-time random field for the Kardar-Parisi-Zhang universality class. While the joint law of the KPZ fixed point at a fixed time has been studied extensively, the multipoint distributions of the KPZ fixed point in the general space-time plane are much less well understood. More explicitly, formulas were only available for the narrow wedge initial condition arXiv:1906.01053, arXiv:1907.09876 and the flat initial condition arXiv:1907.09876 for the multipoint distributions, and the half-Brownian and Brownian initial conditions arXiv:2010.07357v1, arXiv:2504.19975 for the two-point distributions.
In this paper, we obtain the first formula for the space-time joint distributions of the KPZ fixed point with general initial conditions of compact support. The formula is obtained through taking $1:2:3$ KPZ scaling limit of the multipoint distribution formulas for the totally asymmetric simple exclusion process (TASEP). A key ingredient is a probabilistic representation, inspired by arXiv:1701.00018, of the kernel encoding the initial condition for TASEP, which was first defined through an implicit characterization in arXiv:1907.09876. Moreover, we also verify that the equal time version of our formula matches the path integral formula in arXiv:1701.00018 for the KPZ fixed point when the initial condition is of compact support. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_03246 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multipoint distributions of the KPZ fixed point with compactly supported initial conditions Liao, Yuchen Liu, Zhipeng Probability Mathematical Physics 60K35 (Primary), 60G60 (Secondary) The KPZ fixed point is a universal limiting space-time random field for the Kardar-Parisi-Zhang universality class. While the joint law of the KPZ fixed point at a fixed time has been studied extensively, the multipoint distributions of the KPZ fixed point in the general space-time plane are much less well understood. More explicitly, formulas were only available for the narrow wedge initial condition arXiv:1906.01053, arXiv:1907.09876 and the flat initial condition arXiv:1907.09876 for the multipoint distributions, and the half-Brownian and Brownian initial conditions arXiv:2010.07357v1, arXiv:2504.19975 for the two-point distributions. In this paper, we obtain the first formula for the space-time joint distributions of the KPZ fixed point with general initial conditions of compact support. The formula is obtained through taking $1:2:3$ KPZ scaling limit of the multipoint distribution formulas for the totally asymmetric simple exclusion process (TASEP). A key ingredient is a probabilistic representation, inspired by arXiv:1701.00018, of the kernel encoding the initial condition for TASEP, which was first defined through an implicit characterization in arXiv:1907.09876. Moreover, we also verify that the equal time version of our formula matches the path integral formula in arXiv:1701.00018 for the KPZ fixed point when the initial condition is of compact support. |
| title | Multipoint distributions of the KPZ fixed point with compactly supported initial conditions |
| topic | Probability Mathematical Physics 60K35 (Primary), 60G60 (Secondary) |
| url | https://arxiv.org/abs/2509.03246 |