High-dimensional near critical percolation and the torus plateau
Consider bond percolation with probablity $p$ on $Z^d$ where
either $d>10$ or $d>6$ and the model is sufficiently spread-out and let
$\tau_p(x)$ denote the percolation two-point function. It is well known
that there exists a critical value $p_c>0$ such that for $p < p_c$,
$\tau_p(x)$ decays exponentially fast in $|x|$, and at $p_c$
$\tau_{p_c}(x)$ decays like $|x|^{2-d}$. We employ a wide range of
techniques to obtain an upper bound on $\tau_p(x)$ that interpolates
between these two regimes and which, in some sense, is optimal. A
similar result is obtained for the slightly subcritical one-arm
probability. The main ingredients of the proof will be presented: in
particular, we will insist on the notion of pioneer edges on which our
work relies heavily.
As an application, we illustrate how the near-critical decay of
$\tau_p(x)$ on $Z^d$ can be used to prove that throughout the so-called
critical window, the torus two-point function has a plateau, meaning
that it decays for small $x$ like $|x|^{2-d}$ but for large $x$ it is
essentially constant and of order $V^{-2/3}$ where $V$ is the volume
(number of points) of the torus. This plateau estimate leads to a novel
and more direct proof of the torus triangle condition.
This is based on joint work with Tom Hutchcroft and Gordon Slade (link
to the preprint: https://arxiv.org/abs/2107.12971).