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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.06449 |
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| _version_ | 1866909573440339968 |
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| author | Hu, Kevin Ramanan, Kavita |
| author_facet | Hu, Kevin Ramanan, Kavita |
| contents | For any integer $κ\geq 2$, the $κ$-local-field equation ($κ$-LFE) characterizes the limit of the neighborhood path empirical measure of interacting diffusions on $κ$-regular random graphs, as the graph size goes to infinity. It has been conjectured that the long-time behavior of the (in general non-Markovian) $κ$-LFE coincides with that of a certain more tractable Markovian analog, the Markov $κ$-local-field equation. In the present article, we prove this conjecture for the case when $κ= 2$ and the diffusions are one-dimensional with affine drifts. As a by-product of our proof, we also show that for interacting diffusions on the $n$-cycle (or 2-regular random graph on $n$ vertices), the limits $n \rightarrow \infty$ and $t\rightarrow \infty$ commute. Along the way, we also establish well-posedness of the Markov $κ$-local field equations with affine drifts for all $κ\geq 2$, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06449 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A case study of the long-time behavior of the Gaussian local-field equation Hu, Kevin Ramanan, Kavita Probability 60K35, 60J60 For any integer $κ\geq 2$, the $κ$-local-field equation ($κ$-LFE) characterizes the limit of the neighborhood path empirical measure of interacting diffusions on $κ$-regular random graphs, as the graph size goes to infinity. It has been conjectured that the long-time behavior of the (in general non-Markovian) $κ$-LFE coincides with that of a certain more tractable Markovian analog, the Markov $κ$-local-field equation. In the present article, we prove this conjecture for the case when $κ= 2$ and the diffusions are one-dimensional with affine drifts. As a by-product of our proof, we also show that for interacting diffusions on the $n$-cycle (or 2-regular random graph on $n$ vertices), the limits $n \rightarrow \infty$ and $t\rightarrow \infty$ commute. Along the way, we also establish well-posedness of the Markov $κ$-local field equations with affine drifts for all $κ\geq 2$, which may be of independent interest. |
| title | A case study of the long-time behavior of the Gaussian local-field equation |
| topic | Probability 60K35, 60J60 |
| url | https://arxiv.org/abs/2504.06449 |