The discrete analogue of the Gaussian
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913512706539520 |
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| author | Chinta, Gautam Jorgenson, Jay Karlsson, Anders Smajlović, Lejla |
| author_facet | Chinta, Gautam Jorgenson, Jay Karlsson, Anders Smajlović, Lejla |
| contents | This paper illustrates the utility of the heat kernel on $\mathbb{Z}$ as the discrete analogue of the Gaussian density function. It is the two-variable function $K_{\mathbb{Z}}(t,x)=e^{-2t}I_{x}(2t)$ involving a Bessel function and variables $x\in\mathbb{Z}$ and real $t\geq 0$. Like its classic counterpart it appears in many mathematical and physical contexts and has a wealth of applications. Some of these will be reviewed here, concerning Bessel integrals, trigonometric sums, hypergeometric functions and asymptotics of discrete models appearing in statistical and quantum physics. Moreover, we prove a new local limit theorem for sums of integer-valued random variables, obtain novel special values of the spectral zeta function of Bethe lattices, and provide a discussion on how $e^{-2t}I_{x}(2t)$ could be useful in differential privacy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_14344 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The discrete analogue of the Gaussian Chinta, Gautam Jorgenson, Jay Karlsson, Anders Smajlović, Lejla Mathematical Physics Probability 81Q99, 81T25, 35Gxx, 58J35, 60F05, 68P27 This paper illustrates the utility of the heat kernel on $\mathbb{Z}$ as the discrete analogue of the Gaussian density function. It is the two-variable function $K_{\mathbb{Z}}(t,x)=e^{-2t}I_{x}(2t)$ involving a Bessel function and variables $x\in\mathbb{Z}$ and real $t\geq 0$. Like its classic counterpart it appears in many mathematical and physical contexts and has a wealth of applications. Some of these will be reviewed here, concerning Bessel integrals, trigonometric sums, hypergeometric functions and asymptotics of discrete models appearing in statistical and quantum physics. Moreover, we prove a new local limit theorem for sums of integer-valued random variables, obtain novel special values of the spectral zeta function of Bethe lattices, and provide a discussion on how $e^{-2t}I_{x}(2t)$ could be useful in differential privacy. |
| title | The discrete analogue of the Gaussian |
| topic | Mathematical Physics Probability 81Q99, 81T25, 35Gxx, 58J35, 60F05, 68P27 |
| url | https://arxiv.org/abs/2409.14344 |