Dice Question Streamline Icon: https://streamlinehq.com

Asymptotics of p-capacity for m-dimensional balls (m<d)

Determine the asymptotic order (in radius r and dimension parameters) of pCap of m-dimensional balls embedded in Z^d for 1 ≤ m < d, in the high-dimensional critical percolation regime, analogously to known results for branching capacity.

Information Square Streamline Icon: https://streamlinehq.com

Background

The authors fully characterize the order of pCap(B(0,r)) for d-dimensional cubes (balls in sup norm) and provide lower bounds via Bessel–Riesz capacity, but do not treat lower-dimensional geometric subsets.

Extending estimates to m-dimensional balls would parallel existing branching capacity results and clarify how geometry and codimension affect p-capacity.

References

On the other hand, it remains an interesting open problem to estimate the p-capacity of $m$-dimensional balls, with $m<d$, similarly to what is known for branching capacity.

— Capacity in high dimensional percolation (2509.21253 - Asselah et al., 25 Sep 2025) in Introduction, after Theorem \ref{thm.pcap.ball}