Slab percolation for the Ising model revisited
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910423710695424 |
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| author | Severo, Franco |
| author_facet | Severo, Franco |
| contents | In this note, we give a new and short proof for a theorem of Bodineau stating that the slab percolation threshold $\hat{p}_c$ for the FK-Ising model coincides with the standard percolation critical point $p_c$ in all dimensions $d\geq3$. Both proofs rely on the positivity of the surface tension for $p>p_c$ proved by Lebowitz & Pfister. The key difference is that while Bodineau's proof is based on a delicate dynamic renormalization inspired by the work of Barsky, Grimmett & Newman, our proof utilizes a technique of Benjamini & Tassion to prove the uniqueness of macroscopic clusters via sprinkling, which then implies percolation on slabs through a rather straightforward static renormalization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_06831 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Slab percolation for the Ising model revisited Severo, Franco Probability 82B43, 60K35 In this note, we give a new and short proof for a theorem of Bodineau stating that the slab percolation threshold $\hat{p}_c$ for the FK-Ising model coincides with the standard percolation critical point $p_c$ in all dimensions $d\geq3$. Both proofs rely on the positivity of the surface tension for $p>p_c$ proved by Lebowitz & Pfister. The key difference is that while Bodineau's proof is based on a delicate dynamic renormalization inspired by the work of Barsky, Grimmett & Newman, our proof utilizes a technique of Benjamini & Tassion to prove the uniqueness of macroscopic clusters via sprinkling, which then implies percolation on slabs through a rather straightforward static renormalization. |
| title | Slab percolation for the Ising model revisited |
| topic | Probability 82B43, 60K35 |
| url | https://arxiv.org/abs/2312.06831 |