Asymptotics of the finite-temperature sine kernel determinant
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909369671614464 |
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| author | Xu, Shuai-Xia |
| author_facet | Xu, Shuai-Xia |
| contents | In the present paper, we study the asymptotics of the Fredholm determinant $D(x,s)$ of the finite-temperature deformation of the sine kernel, which represents the probability that there is no particles on the interval $(-x/π,x/π)$ in the bulk scaling limit of the finite-temperature fermion system. The variable $s$ in $D(x,s)$ is related to the temperature. The determinant also corresponds to the finite-temperature correlation function of one dimensional Bose gas. We derive the asymptotics of $D(x,s)$ in several different regimes in the $(x,s)$-plane. A third-order phase transition is observed in the asymptotic expansions as both $x$ and $s$ tend to positive infinity at certain related speed. The phase transition is then shown to be described by an integral involving the Hastings-McLeod solution of the second Painlevé equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06722 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotics of the finite-temperature sine kernel determinant Xu, Shuai-Xia Mathematical Physics In the present paper, we study the asymptotics of the Fredholm determinant $D(x,s)$ of the finite-temperature deformation of the sine kernel, which represents the probability that there is no particles on the interval $(-x/π,x/π)$ in the bulk scaling limit of the finite-temperature fermion system. The variable $s$ in $D(x,s)$ is related to the temperature. The determinant also corresponds to the finite-temperature correlation function of one dimensional Bose gas. We derive the asymptotics of $D(x,s)$ in several different regimes in the $(x,s)$-plane. A third-order phase transition is observed in the asymptotic expansions as both $x$ and $s$ tend to positive infinity at certain related speed. The phase transition is then shown to be described by an integral involving the Hastings-McLeod solution of the second Painlevé equation. |
| title | Asymptotics of the finite-temperature sine kernel determinant |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2403.06722 |