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Boundary small-ball hit probability scaling for restricted clusters

Establish that for critical percolation on Z^d in the high-dimensional regime and for all r,s≥1 and all x∈∂B(0,r), the probability that the restricted cluster C_r(0) intersects the boundary cap Q_s(x) satisfies P( C_r(0) ∩ Q_s(x) ≠ ∅ ) ≲ s^{d−3}/r^{d−1}, uniformly in x, where C_r(0) is the cluster of 0 using edges lying in B(0,r) and Q_s(x) = B(x,s) ∩ ∂B(0,r).

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Background

The main large-deviation bound in Theorem 4 features a prefactor 1/r2; the authors conjecture a sharper prefactor that matches expected boundary behavior.

Proving this scaling would refine pioneer-point large deviations and strengthen tools for regularity-based arguments in high-dimensional percolation.

References

However, we conjecture that the probability for $\mathcal C_r(0)$ to intersect $Q_s(x)$ is at most of order $s{d-3}/r{d-1}$, uniformly in $x\in \partial B(0,r).

— Capacity in high dimensional percolation (2509.21253 - Asselah et al., 25 Sep 2025) in Section 4 (Large deviations), discussion after Theorem \ref{thm.LD}