Two-set product limit for p-capacity under IIC normalization

Establish that for critical Bernoulli percolation on Z^d in the high-dimensional regime (nearest-neighbor d>10 or sufficiently spread-out d>6), and for any finite sets A,B ⊂ Z^d, the limit lim_{||z||→∞} [ pCap(A) + pCap(B) − pCap( A ∪ (z + B) ) ] / τ_∞(z) equals pCap(A)·pCap(B), where τ_∞(z) = P( z ∈ C_∞(0) ).

Background

Building on Conjecture 1, the authors formulate a stronger two-set version mirroring product structures known for branching capacity and Bessel–Riesz capacities.

This conjecture would characterize the asymptotic interaction between two distant finite sets A and z+B through the IIC, yielding a factorization by their p-capacities.