On the multipoint distribution formulas of the parabolic Airy process
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908469329657856 |
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| author | Liu, Zhipeng Ortiz, Aaron |
| author_facet | Liu, Zhipeng Ortiz, Aaron |
| contents | The parabolic Airy process is the Airy$_2$ process minus a parabola, initially defined by its finite-dimensional distributions, which are given by a Fredholm determinant formula with the extended Airy kernel. This process is also the one-time spatial marginal of the KPZ fixed point with the narrow wedge initial condition. There are two formulas for the space-time multipoint distribution of the KPZ fixed point with the narrow wedge initial condition obtained by arXiv:1906.01053 and arXiv:1907.09876. Especially, the equal-time case of arXiv:1907.09876 gives a different formula of the multipoint distribution of the parabolic Airy process. In this paper, we present a direct proof that this formula matches the one with the extended Airy kernel. Some byproducts in the proof include several new formulas for the parabolic Airy process, and a generalization of the Andreief's identity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20487 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the multipoint distribution formulas of the parabolic Airy process Liu, Zhipeng Ortiz, Aaron Probability The parabolic Airy process is the Airy$_2$ process minus a parabola, initially defined by its finite-dimensional distributions, which are given by a Fredholm determinant formula with the extended Airy kernel. This process is also the one-time spatial marginal of the KPZ fixed point with the narrow wedge initial condition. There are two formulas for the space-time multipoint distribution of the KPZ fixed point with the narrow wedge initial condition obtained by arXiv:1906.01053 and arXiv:1907.09876. Especially, the equal-time case of arXiv:1907.09876 gives a different formula of the multipoint distribution of the parabolic Airy process. In this paper, we present a direct proof that this formula matches the one with the extended Airy kernel. Some byproducts in the proof include several new formulas for the parabolic Airy process, and a generalization of the Andreief's identity. |
| title | On the multipoint distribution formulas of the parabolic Airy process |
| topic | Probability |
| url | https://arxiv.org/abs/2507.20487 |