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

TL;DR

This paper investigates the asymptotic behavior of the $p$-capacity in a lattice as the scale grows, revealing a logarithmic decay rate at the critical dimension where $p=d$, with implications for large deviations in first passage percolation.

Contribution

It provides the first precise asymptotic formula for the $p$-capacity at the critical case $p=d$, including an explicit constant, filling a gap in the understanding of capacity behavior.

Findings

For $p<d$, the capacity converges to a positive constant.

For $p>d$, the capacity vanishes polynomially fast.

At $p=d$, the capacity vanishes as $c_d (rac{1}{ ext{log} n})^{d-1}$ with an explicit constant.

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 $f\geq 1$ 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 $c_d$.