IIC hitting probability normalized by τ∞ equals p-capacity

Establish that for critical Bernoulli percolation on Z^d in the high-dimensional (mean-field) regime where the two-point function satisfies τ(z) ∼ a_d·||z||^{2−d} (nearest-neighbor for d>10 or sufficiently spread-out for d>6), the limit lim_{||z||→∞} P( C_∞(z) ∩ A ≠ ∅ ) / τ_∞(z) equals pCap(A) for every finite set A ⊂ Z^d, where τ_∞(z) = P( z ∈ C_∞(0) ) and pCap(A) = lim_{||z||→∞} P(z ↔ A)/τ(z).

Background

The paper introduces p-capacity pCap(A) via the asymptotic probability that a distant cluster intersects a finite set, and proves upper and lower bounds relating IIC hitting probabilities to pCap(A). The authors expect a precise limiting relation mirroring analogous results for branching random walks.

Conjecture 1 proposes that the normalized IIC hitting probability converges to pCap(A), strengthening Theorem 1.2 which gives matching upper and lower bounds up to constants. This would parallel Zhu’s branching capacity theory.