High-dimensional near-critical percolation and the torus plateau. (English) Zbl 1532.60216

Hutchcroft, Tom; Michta, Emmanuel; Slade, Gordon

Ann. Probab. 51, No. 2, 580-625 (2023).

Summary: We consider percolation on \({\mathbb{Z}^d}\) and on the \(d\)-dimensional discrete torus, in dimensions \(d\ge 11\) for the nearest-neighbour model and in dimensions \(d > 6\) for spread-out models. For \({\mathbb{Z}^d}\) 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 \[ \begin{aligned} \mathbb{P}_{p_{c} -\varepsilon} (0\leftrightarrow x) &\le \frac{C}{\| x{\|^{d-2}}}{e^{-c{\varepsilon^{1/2}}\| x\|}},\\ \frac{c}{{r^2}}{e^{-C{\varepsilon^{1/2}}r}} &\le {\mathbb{P}_{{p_c}-\varepsilon}}(0\leftrightarrow \partial{[-r,r]^d})\le \frac{C}{{r^2}}{e^{-c{\varepsilon^{1/2}}r}}. \end{aligned} \] 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 \({\mathbb{Z}^d}\) 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 \({\mathbb{Z}^d}\) two-point function. In particular, we use results derived from the lace expansion on \({\mathbb{Z}^d}\), but in contrast to previous work on high-dimensional torus percolation, we do not need or use a separate torus lace expansion.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
82B27	Critical phenomena in equilibrium statistical mechanics
05C80	Random graphs (graph-theoretic aspects)
82B43	Percolation

Keywords:

lace expansion; one-arm exponent; percolation; torus plateau; triangle condition; two-point function

References:

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