On cable-graph percolation between dimensions 2 and 3
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| Format: | Preprint |
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2025
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| _version_ | 1866915657111568384 |
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| author | Rodriguez, Pierre-François Zhang, Wen |
| author_facet | Rodriguez, Pierre-François Zhang, Wen |
| contents | We consider the Gaussian free field on two-dimensional slabs with a thickness described by a height $h$ at spatial scale $N$. We investigate the radius of critical clusters for the associated cable-graph percolation problem, which depends sensitively on the parameter $h$. Our results unveil a whole family of new "fixed points", which interpolate between recent results from arXiv:2303.03782 in two dimensions and from arXiv:2405.17417 and arXiv:2406.02397 in three dimensions, and describe critical behaviour beyond those regimes. In the delocalised phase, the one-arm decay exhibits a "plateau", i.e. it doesn't depend on the speed at which the variance of the field diverges in the large-$N$ limit. Our methods rely on a careful analysis of the interplay between two- and three-dimensional effects for the underlying random walk, which manifest themselves in a corresponding decomposition of the field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_05947 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On cable-graph percolation between dimensions 2 and 3 Rodriguez, Pierre-François Zhang, Wen Probability Mathematical Physics 60K35, 60G15, 60J45, 82B43 We consider the Gaussian free field on two-dimensional slabs with a thickness described by a height $h$ at spatial scale $N$. We investigate the radius of critical clusters for the associated cable-graph percolation problem, which depends sensitively on the parameter $h$. Our results unveil a whole family of new "fixed points", which interpolate between recent results from arXiv:2303.03782 in two dimensions and from arXiv:2405.17417 and arXiv:2406.02397 in three dimensions, and describe critical behaviour beyond those regimes. In the delocalised phase, the one-arm decay exhibits a "plateau", i.e. it doesn't depend on the speed at which the variance of the field diverges in the large-$N$ limit. Our methods rely on a careful analysis of the interplay between two- and three-dimensional effects for the underlying random walk, which manifest themselves in a corresponding decomposition of the field. |
| title | On cable-graph percolation between dimensions 2 and 3 |
| topic | Probability Mathematical Physics 60K35, 60G15, 60J45, 82B43 |
| url | https://arxiv.org/abs/2512.05947 |