Spectral dimensions for one-dimensional critical long-range percolation
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908395827625984 |
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| author | Fan, Zherui Huang, Lu-Jing |
| author_facet | Fan, Zherui Huang, Lu-Jing |
| contents | Consider the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-β\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}d ud v\}$ for $|i-j|>1$ for some fixed $β>0$ and with probability 1 for $|i-j|=1$. We prove that both the quenched and annealed spectral dimensions of the associated simple random walk are $2/(1+δ)$, where $δ\in (0,1)$ is the exponent of the effective resistance in the LRP model, as derived in [10, Theorem 1.1]. Our work addresses an open question from [7, Section 5]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_15037 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral dimensions for one-dimensional critical long-range percolation Fan, Zherui Huang, Lu-Jing Probability 60K35, 82B27, 82B43 Consider the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-β\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}d ud v\}$ for $|i-j|>1$ for some fixed $β>0$ and with probability 1 for $|i-j|=1$. We prove that both the quenched and annealed spectral dimensions of the associated simple random walk are $2/(1+δ)$, where $δ\in (0,1)$ is the exponent of the effective resistance in the LRP model, as derived in [10, Theorem 1.1]. Our work addresses an open question from [7, Section 5]. |
| title | Spectral dimensions for one-dimensional critical long-range percolation |
| topic | Probability 60K35, 82B27, 82B43 |
| url | https://arxiv.org/abs/2505.15037 |