References & Citations
Mathematics > Probability
Title: Slab percolation for the Ising model revisited
(Submitted on 11 Dec 2023 (v1), last revised 26 Apr 2024 (this version, v2))
Abstract: 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.
Submission history
From: Franco Severo [view email][v1] Mon, 11 Dec 2023 20:43:23 GMT (13kb)
[v2] Fri, 26 Apr 2024 12:04:19 GMT (45kb,D)
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