Subcritical Connectivity and Some Exact Tail Exponents in High Dimensional Percolation

Abstract

In high dimensional percolation at parameter p < p c , the one-arm probability π p (n) is known to decay exponentially on scale (p c − p) −1/2 . We show the same statement for the ratio π p (n)/π pc (n), establishing a form of a hypothesis of scaling theory.As part of our study, we provide sharp estimates (with matching upper and lower bounds) for several quantities of interest at the critical probability p c . These include the tail behavior of volumes of, and chemical distances within, spanning clusters, along with the scaling of the two-point function at "mesoscopic distance" from the boundary of half-spaces. As a corollary, we obtain the tightness of the number of spanning clusters of a diameter n box on scale n d−6 ; this result complements a lower bound of Aizenman [1].

Slightly supercritical percolation on non‐amenable graphs I: The distribution of finite clusters

“…Kozma and Nachmias [45] then applied this result to prove the 𝑝 = 𝑝 𝑐 case of (5.3). The subcritical case of (5.3) has very recently been established in the independent works [15,40], with [40] also establishing the upper bound of (5.4) in the subcritical case. In contrast, almost no progress has been made on the slightly supercritical cases of these conjectures.…”

Slightly supercritical percolation on nonamenable graphs II: Growth and isoperimetry of infinite clusters

“…The critical version of this estimate, stating that E pc #B int (v, r) ≍ r for every r ≥ 1, was proven to hold for any transitive graph satisfying the triangle condition (which is implied by the L 2 boundedness condition [21, p.4]) in [27,32]. The transition from critical-like to supercritical-like behaviour outside a scaling window of intrinsic radius |p − p c | −1 is typical of off-critical percolation in high-dimensional settings [7,23,24]. As is common to such analyses, our proofs will often treat the inside-window and outside-window cases separately, with the inside-window results following straightforwardly from what is known about critical percolation.…”

Mean-field bounds for Poisson-Boolean percolation

“…and under the standard percolation near-critical scaling hypothesis (see [12,Chapter 9]) it is natural to expect that this bound is sharp, up to constants in the exponent, for d > d c = 6 (this has recently been proven in sufficiently high dimension [16], and see also [4,18] for related 'near-critical scaling' results for Bernoulli percolation in sufficiently high dimension).…”