E. Michta & G. Slade: High-dimensional near-critical percolation and the torus plateau

Abstract: We consider percolation on \mathbb{Z}^d at and near the critical point in high dimensions (d>6 for spread-out models or d>10 for the nearest-neighbour model), where the two-point function has been proved to have power law decay at the critical point and exponential decay below the critical point. We obtain an upper bound on the two-point function that interpolates between these two regimes and which is essentially optimal. A similar result is obtained for the slightly subcritical one-arm probability. As an application, we use the near-critical decay of the two-point function to prove that throughout the critical window for percolation on a high-dimensional torus, the torus two-point function has a plateau: it decays at small distances with the same power as on \mathbb{Z}^d but at larger distances is essentially constant and of order V^{−2/3} where V is the volume of the torus. This plateau estimate leads to a new and direct proof of the torus triangle condition (which has many consequences for critical torus percolation) using \mathbb{Z}^d results, without any torus lace expansion.

The main ingredients of the proof will be presented. These include previous results for high-dimensional \mathbb{Z}^d percolation, the notion of pioneer edges, OSSS inequality and diagrammatic estimates.

The talk is based on joint work with Tom Hutchcroft and the two speakers https://arxiv.org/abs/2107.12971.

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