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Journal of Convex Analysis 33 (2026), No. 1&2, 013--027
Copyright Heldermann Verlag 2026

Asymptotics of the p-Capacity in the Critical Regime

Clément Cosco
Université Dauphine, Paris, France
cosco@ceremade.dauphine.fr

Shuta Nakajima
Dept. of Mathematics, Meiji University, Tokyo, Japan
njima@meiji.ac.jp

Florian Schweiger
(1) Weizmann Institute of Science, Rechovot, Israel
(2) Section de Mathématiques, Université de Genève, Switzerland
florian.schweiger@unige.ch


[Abstract-pdf]

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 $pd$ 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.

Keywords: p-Capacity, variational problem, first passage percolation.

MSC: 31C45; 31C20, 94C15, 60K35.

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