Gaussian Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé IV System
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916460179226624 |
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| author | Chen, Yang Lyu, Shulin |
| author_facet | Chen, Yang Lyu, Shulin |
| contents | We study the Hankel determinant generated by the Gaussian weight with jump discontinuities at $t_1,\cdots,t_m$. By making use of a pair of ladder operators satisfied by the associated monic orthogonal polynomials and three supplementary conditions, we show that the logarithmic derivative of the Hankel determinant satisfies a second order partial differential equation which is reduced to the $σ$-form of a Painlevé IV equation when $m=1$. Moreover, under the assumption that $t_k-t_1$ is fixed for $k=2,\cdots,m$, by considering the Riemann-Hilbert problem for the orthogonal polynomials, we construct direct relationships between the auxiliary quantities introduced in the ladder operators and solutions of a coupled Painlevé IV system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_04127 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Gaussian Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé IV System Chen, Yang Lyu, Shulin Mathematical Physics 15B52, 33E17, 42C05 We study the Hankel determinant generated by the Gaussian weight with jump discontinuities at $t_1,\cdots,t_m$. By making use of a pair of ladder operators satisfied by the associated monic orthogonal polynomials and three supplementary conditions, we show that the logarithmic derivative of the Hankel determinant satisfies a second order partial differential equation which is reduced to the $σ$-form of a Painlevé IV equation when $m=1$. Moreover, under the assumption that $t_k-t_1$ is fixed for $k=2,\cdots,m$, by considering the Riemann-Hilbert problem for the orthogonal polynomials, we construct direct relationships between the auxiliary quantities introduced in the ladder operators and solutions of a coupled Painlevé IV system. |
| title | Gaussian Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé IV System |
| topic | Mathematical Physics 15B52, 33E17, 42C05 |
| url | https://arxiv.org/abs/2304.04127 |