Slab percolation for the Ising model revisited

2023/12/11 by Severo, Franco
#60K35 #82B43 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2312.06831

Abstract

In this note, we give a new and short proof for a theorem of Bodineau stating that the slab percolation threshold p_c for the FK-Ising model coincides with the standard percolation critical point p_c in all dimensions d≥3. 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.

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