Limiting distribution of the chemical distance in high dimensional critical percolation
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915612561768448 |
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| author | Chatterjee, Shirshendu Chinmay, Pranav Hanson, Jack Sosoe, Philippe |
| author_facet | Chatterjee, Shirshendu Chinmay, Pranav Hanson, Jack Sosoe, Philippe |
| contents | We identify the asymptotic distribution of the chemical distance in high-dimensional critical Bernoulli percolation. Namely, we show that the distance between the origin and a distant vertex conditioned to lie in the cluster of the origin converges in distribution when rescaled by a multiple the square of the Euclidean distance. The limiting distribution has an explicit density and coincides with the distribution of the time for a Brownian motion in $\mathbb{R}^d$ conditioned to hit a given unit vector to reach its target. Our result follows from a general moment computation for quantities that have an additive structure across the pivotal edges on a long-range connection in percolation. In addition to the number of pivotal edges in a long connection, this also includes the effective resistance. The existence of the incipient infinite cluster limit, in a form recently established, plays a key role in the derivation of our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06236 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Limiting distribution of the chemical distance in high dimensional critical percolation Chatterjee, Shirshendu Chinmay, Pranav Hanson, Jack Sosoe, Philippe Probability Mathematical Physics 60K35, 82B43 We identify the asymptotic distribution of the chemical distance in high-dimensional critical Bernoulli percolation. Namely, we show that the distance between the origin and a distant vertex conditioned to lie in the cluster of the origin converges in distribution when rescaled by a multiple the square of the Euclidean distance. The limiting distribution has an explicit density and coincides with the distribution of the time for a Brownian motion in $\mathbb{R}^d$ conditioned to hit a given unit vector to reach its target. Our result follows from a general moment computation for quantities that have an additive structure across the pivotal edges on a long-range connection in percolation. In addition to the number of pivotal edges in a long connection, this also includes the effective resistance. The existence of the incipient infinite cluster limit, in a form recently established, plays a key role in the derivation of our results. |
| title | Limiting distribution of the chemical distance in high dimensional critical percolation |
| topic | Probability Mathematical Physics 60K35, 82B43 |
| url | https://arxiv.org/abs/2509.06236 |