Asymptotics of the p-Capacity in the Critical Regime

We are interested in the asymptotics of the

-capacity between the origin and the set

nB n B

, where

B B

is the boundary of the unit ball of the lattice

\mathbb Z^d Z d

. The

p p

-capacity is defined as the minimum of the Dirichlet energy associated with a discrete version of the

p p

-Laplacian. This variational problem has arisen in particular in the study of large deviations for first passage percolation. For

p<d p < d

, the

p p

-capacity converges to some positive constant, while for

p>d p > d

the capacity vanishes polynomially fast. The present paper deals with the case

p=d p = d

, for which we prove that the

p p

-capacity vanishes as

c_d (\log n)^{-d+1} c d ( log ⁡ n ) − d + 1

with an explicit constant

c_d c d

. Our proof relies on Thomson's principle for the p-capacity.