$L^2$-cutoff for the averaging process on random regular graphs
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2026
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| author | Caputo, Pietro Quattropani, Matteo Sau, Federico |
| author_facet | Caputo, Pietro Quattropani, Matteo Sau, Federico |
| contents | We study the mixing time of the averaging process on a large random $d$-regular graph, $d\ge 3$, and prove an $L^2$-cutoff with an explicit cutoff time. Somewhat surprisingly, we uncover a phase transition at the finite, fixed degree $d=10$: for small degrees, i.e., $d\le 10$, the averaging process mixes as fast as the corresponding random walk on the same graph, whereas for $d> 10$ its $L^2$-mixing is governed by a different, slower mechanism. Our proof relies on a detailed asymptotic analysis of an auxiliary biased birth-and-death chain with a slow bond. We also briefly discuss an analogous phase transition for the $L^1$-mixing. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00705 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $L^2$-cutoff for the averaging process on random regular graphs Caputo, Pietro Quattropani, Matteo Sau, Federico Probability 60K35 We study the mixing time of the averaging process on a large random $d$-regular graph, $d\ge 3$, and prove an $L^2$-cutoff with an explicit cutoff time. Somewhat surprisingly, we uncover a phase transition at the finite, fixed degree $d=10$: for small degrees, i.e., $d\le 10$, the averaging process mixes as fast as the corresponding random walk on the same graph, whereas for $d> 10$ its $L^2$-mixing is governed by a different, slower mechanism. Our proof relies on a detailed asymptotic analysis of an auxiliary biased birth-and-death chain with a slow bond. We also briefly discuss an analogous phase transition for the $L^1$-mixing. |
| title | $L^2$-cutoff for the averaging process on random regular graphs |
| topic | Probability 60K35 |
| url | https://arxiv.org/abs/2603.00705 |