Counting minimal cutsets and $p_c<1$
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917012498808832 |
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| author | Easo, Philip Severo, Franco Tassion, Vincent |
| author_facet | Easo, Philip Severo, Franco Tassion, Vincent |
| contents | We prove two results concerning percolation on general graphs.
- We establish the converse of the classical Peierls argument: if the critical parameter for (uniform) percolation satisfies $p_c<1$, then the number of minimal cutsets of size $n$ separating a given vertex from infinity is bounded above exponentially in $n$. This resolves a conjecture of Babson and Benjamini from 1999.
- We prove that $p_c<1$ for every uniformly transient graph. This solves a problem raised by Duminil-Copin, Goswami, Raoufi, Severo and Yadin, and provides a new proof that $p_c<1$ for every transitive graph of superlinear growth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04539 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting minimal cutsets and $p_c<1$ Easo, Philip Severo, Franco Tassion, Vincent Probability Mathematical Physics Combinatorics Group Theory 82B43, 60K35, 05C81, 05C70 We prove two results concerning percolation on general graphs. - We establish the converse of the classical Peierls argument: if the critical parameter for (uniform) percolation satisfies $p_c<1$, then the number of minimal cutsets of size $n$ separating a given vertex from infinity is bounded above exponentially in $n$. This resolves a conjecture of Babson and Benjamini from 1999. - We prove that $p_c<1$ for every uniformly transient graph. This solves a problem raised by Duminil-Copin, Goswami, Raoufi, Severo and Yadin, and provides a new proof that $p_c<1$ for every transitive graph of superlinear growth. |
| title | Counting minimal cutsets and $p_c<1$ |
| topic | Probability Mathematical Physics Combinatorics Group Theory 82B43, 60K35, 05C81, 05C70 |
| url | https://arxiv.org/abs/2412.04539 |