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Variational characterization of p-capacity

Develop a variational formula for pCap(A) of a finite set A ⊂ Z^d in high-dimensional critical percolation, analogous to the Bessel–Riesz capacity variational formula, and ascertain whether minimizers (equilibrium measures) exist for this characterization.

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Background

While the paper introduces equilibrium-measure representations and analogies to random-walk reversibility, a direct variational characterization of p-capacity is not established.

A variational formula would provide structural insight into p-capacity and facilitate comparisons to classical potentials, capacities, and branching capacity.

References

However, we were not able to prove any variational formula for $\textrm{pCap}$, and this remains an important open problem.

— Capacity in high dimensional percolation (2509.21253 - Asselah et al., 25 Sep 2025) in Introduction, paragraph preceding Proposition \ref{prop:eqmeasure}