Asymptotics of the p-Capacity in the Critical Regime

We are interested in the asymptotics of the $p$-capacity between the origin and the set $nB$, where $B$ is the boundary of the unit ball of the lattice $\mathbb Z^d$. The $p$-capacity is defined as the minimum of the Dirichlet energy associated with a discrete version of the $p$-Laplacian. This variational problem has arisen in particular in the study of large deviations for first passage percolation. For $p<d$, the $p$-capacity converges to some positive constant, while for $p>d$ the capacity vanishes polynomially fast. The present paper deals with the case $p=d$, for which we prove that the $p$-capacity vanishes as $c_d (\log n)^{-d+1}$ with an explicit constant $c_d$. Our proof relies on Thomson's principle for the p-capacity.