The scaling limit of random walk and the intrinsic metric on planar critical percolation
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2026
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| _version_ | 1866918448971382784 |
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| author | Đanković, Irina Markering, Maarten Miller, Jason Yuan, Yizheng |
| author_facet | Đanković, Irina Markering, Maarten Miller, Jason Yuan, Yizheng |
| contents | We consider critical site percolation ($p=p_c=1/2$) on the triangular lattice $\mathbf{T}$ in two dimensions. We show that the simple random walk on the clusters of open vertices converges in the scaling limit to a continuous diffusion which lives in the gasket of a conformal loop ensemble with parameter $κ= 6$ $\big(\mathrm{CLE}_6\big)$, the so-called $\mathrm{CLE}_6$ Brownian motion. We also show that the intrinsic (i.e., chemical distance) metric converges in the scaling limit to the geodesic $\mathrm{CLE}_6$ metric. As a consequence, we deduce the existence of the chemical distance exponent, the resistance exponent, and the spectral dimension of the critical percolation clusters. Moreover, we show that the exponents satisfy the Einstein relations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14122 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The scaling limit of random walk and the intrinsic metric on planar critical percolation Đanković, Irina Markering, Maarten Miller, Jason Yuan, Yizheng Probability Mathematical Physics We consider critical site percolation ($p=p_c=1/2$) on the triangular lattice $\mathbf{T}$ in two dimensions. We show that the simple random walk on the clusters of open vertices converges in the scaling limit to a continuous diffusion which lives in the gasket of a conformal loop ensemble with parameter $κ= 6$ $\big(\mathrm{CLE}_6\big)$, the so-called $\mathrm{CLE}_6$ Brownian motion. We also show that the intrinsic (i.e., chemical distance) metric converges in the scaling limit to the geodesic $\mathrm{CLE}_6$ metric. As a consequence, we deduce the existence of the chemical distance exponent, the resistance exponent, and the spectral dimension of the critical percolation clusters. Moreover, we show that the exponents satisfy the Einstein relations. |
| title | The scaling limit of random walk and the intrinsic metric on planar critical percolation |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2604.14122 |