Asymptotics of the $p$-capacity in the critical regime

Abstract

In this note, we are interested in the asymptotics as $n\to \infty$ 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 $\frac{1}{2}{\sum }_{x\in {\mathbb{Z}}^d}{\sum }_{y\sim x}|f(x)-f(y){|}^p$ with $f$ subject to the boundary conditions $f(0)=0$ and $f\ge 1$ on $nB$. 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 $c_d$.

arXiv comment14 pages