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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2506.07377 |
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| _version_ | 1866908399266955264 |
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| author | Prasain, Prem Raj |
| author_facet | Prasain, Prem Raj |
| contents | In this paper, we establish the theory of $p$-modulus of a family of infinite paths on an infinite-rooted tree and then explore its interpretation and properties. One key result is the formulation of $p$-modulus on the infinite tree as a limit of $p$-modulus on truncated trees, with a formula given in terms of a series. Analogous to the existing theory for finite graphs, the $1$-modulus of a family of descending paths in an infinite tree is related to the minimum cut problem, the $2$-modulus is related to effective resistance, and the $\infty$-modulus is related to the length of shortest paths. Another key result is the existence of a critical $p$-value for radially symmetric infinite binary trees, which assigns a kind of dimension to the boundaries |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07377 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $p$-Modulus on radially symmetric trees Prasain, Prem Raj Combinatorics In this paper, we establish the theory of $p$-modulus of a family of infinite paths on an infinite-rooted tree and then explore its interpretation and properties. One key result is the formulation of $p$-modulus on the infinite tree as a limit of $p$-modulus on truncated trees, with a formula given in terms of a series. Analogous to the existing theory for finite graphs, the $1$-modulus of a family of descending paths in an infinite tree is related to the minimum cut problem, the $2$-modulus is related to effective resistance, and the $\infty$-modulus is related to the length of shortest paths. Another key result is the existence of a critical $p$-value for radially symmetric infinite binary trees, which assigns a kind of dimension to the boundaries |
| title | $p$-Modulus on radially symmetric trees |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2506.07377 |