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| Main Authors: | , |
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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2207.09305 |
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| _version_ | 1866917581565198336 |
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| author | Nachmias, Asaf Tang, Pengfei |
| author_facet | Nachmias, Asaf Tang, Pengfei |
| contents | We study the spectral and diffusive properties of the wired minimal spanning forest (WMSF) on the Poisson-weighted infinite tree (PWIT). Let $M$ be the tree containing the root in the WMSF on the PWIT and $(Y_n)_{n\geq0}$ be a simple random walk on $M$ starting from the root. We show that almost surely $M$ has $\mathbb{P}[Y_{2n}=Y_0]=n^{-3/4+o(1)}$ and $\mathrm{dist}(Y_0,Y_n)=n^{1/4+o(1)}$ with high probability. That is, the spectral dimension of $M$ is $\frac{3}{2}$ and its typical displacement exponent is $\frac{1}{4}$, almost surely. These confirm Addario-Berry's predictions in arXiv:1301.1667. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_09305 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The wired minimal spanning forest on the Poisson-weighted infinite tree Nachmias, Asaf Tang, Pengfei Probability We study the spectral and diffusive properties of the wired minimal spanning forest (WMSF) on the Poisson-weighted infinite tree (PWIT). Let $M$ be the tree containing the root in the WMSF on the PWIT and $(Y_n)_{n\geq0}$ be a simple random walk on $M$ starting from the root. We show that almost surely $M$ has $\mathbb{P}[Y_{2n}=Y_0]=n^{-3/4+o(1)}$ and $\mathrm{dist}(Y_0,Y_n)=n^{1/4+o(1)}$ with high probability. That is, the spectral dimension of $M$ is $\frac{3}{2}$ and its typical displacement exponent is $\frac{1}{4}$, almost surely. These confirm Addario-Berry's predictions in arXiv:1301.1667. |
| title | The wired minimal spanning forest on the Poisson-weighted infinite tree |
| topic | Probability |
| url | https://arxiv.org/abs/2207.09305 |