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| Format: | Preprint |
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2024
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| Online adgang: | https://arxiv.org/abs/2403.16475 |
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| _version_ | 1866909188956880896 |
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| author | Dai, Dan Yao, Luming Zhai, Yu |
| author_facet | Dai, Dan Yao, Luming Zhai, Yu |
| contents | In this paper, we investigate a determinantal point process on the interval $(-s,s)$, associated with the confluent hypergeometric kernel. Let $\mathcal{K}^{(α,β)}_s$ denote the trace class integral operator acting on $L^2(-s, s)$ with the confluent hypergeometric kernel. Our focus is on deriving the asymptotics of the Fredholm determinant $\det(I-γ\mathcal{K}^{(α,β)}_s)$ as $s \to +\infty$, while simultaneously $γ\to 1^-$ in a super-exponential region. In this regime of double scaling limit, our asymptotic result also gives us asymptotics of the eigenvalues $λ^{(α, β)}_k(s)$ of the integral operator $\mathcal{K}^{(α,β)}_s$ as $s \to +\infty$. Based on the integrable structure of the confluent hypergeometric kernel, we derive our asymptotic results by applying the Deift-Zhou nonlinear steepest descent method to analyze the related Riemann-Hilbert problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_16475 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotics of the confluent hypergeometric process with a varying external potential in the super-exponential region Dai, Dan Yao, Luming Zhai, Yu Probability Mathematical Physics Classical Analysis and ODEs 33C10, 34M50, 82B26, 45C05 In this paper, we investigate a determinantal point process on the interval $(-s,s)$, associated with the confluent hypergeometric kernel. Let $\mathcal{K}^{(α,β)}_s$ denote the trace class integral operator acting on $L^2(-s, s)$ with the confluent hypergeometric kernel. Our focus is on deriving the asymptotics of the Fredholm determinant $\det(I-γ\mathcal{K}^{(α,β)}_s)$ as $s \to +\infty$, while simultaneously $γ\to 1^-$ in a super-exponential region. In this regime of double scaling limit, our asymptotic result also gives us asymptotics of the eigenvalues $λ^{(α, β)}_k(s)$ of the integral operator $\mathcal{K}^{(α,β)}_s$ as $s \to +\infty$. Based on the integrable structure of the confluent hypergeometric kernel, we derive our asymptotic results by applying the Deift-Zhou nonlinear steepest descent method to analyze the related Riemann-Hilbert problem. |
| title | Asymptotics of the confluent hypergeometric process with a varying external potential in the super-exponential region |
| topic | Probability Mathematical Physics Classical Analysis and ODEs 33C10, 34M50, 82B26, 45C05 |
| url | https://arxiv.org/abs/2403.16475 |