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Optimal s^2 exponential scale for the size of pioneer intersections

Prove that the exponential tail in the large-deviation bound for the number of points of C_r(0) in Q_s(x) has optimal s^2 scaling; specifically, show that conditional on C_r(0)∩Q_s(x) ≠ ∅, there exists c′>0 such that P( |C_r(0)∩Q_s(x)| ≥ t ) ≥ P( C_r(0)∩Q_s(x) ≠ ∅ ) · exp( − c′ t / s^2 ).

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Background

Theorem 4 provides an upper tail bound of exp(−ct/s2) for |C_r(0)∩Q_s(x)| with a 1/r2 prefactor, and the authors conjecture the s2 scale is optimal by proposing a matching lower tail.

Establishing the lower bound would confirm the typical cluster size within boundary caps and sharpen the large-deviation theory for pioneer points.

References

Lastly, we conjecture that the factor $s2$ appearing in the exponential is optimal, in the sense that conditionally on hitting $Q_s(x)$, the size of the cluster $\mathcal C_r(0)$ in $Q_s(x)$ should be typically of order $s2.

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