High-dimensional near-critical percolation and the torus plateau

Abstract

We consider percolation on Zᵈ and on the d-dimensional discrete torus, in dimensions d ≥ 11 for the nearest-neighbour model and in dimensions d > 6 for spread-out models. For Zᵈ we employ a wide range of techniques and previous results to prove that there exist positive constants c and C such that the slightly subcritical two-point function and one-arm probabilities satisfy P_(p_c − ε)(0 ↔ x) ≤ C/(∥x∥ᵈ⁻²)e^(−cε^(1/2)∥x∥), (c/r²)e^(−Cε^((1/2)r)) ≤ P_(pc−ε)(0 ↔ ∂[−r,r]ᵈ) ≤ C/(r²)e^(−cε(1/2)r). Using this, we prove that throughout the critical window the torus two-point function has a "plateau," meaning that it decays for small x as ∥x∥⁻⁽ᵈ⁻²⁾ but for large x is essentially constant and of order V^(−2/3) where V is the volume of the torus. The plateau for the two-point function leads immediately to a proof of the torus triangle condition, which is known to have many implications for the critical behaviour on the torus, and also leads to a proof that the critical values on the torus and on Zᵈ are separated by a multiple of V^(−1/3). The torus triangle condition and the size of the separation of critical points have been proved previously, but our proofs are different and are direct consequences of the bound on the Zᵈ two-point function. In particular, we use results derived from the lace expansion on Zᵈ, but in contrast to previous work on high-dimensional torus percolation, we do not need or use a separate torus lace expansion.

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