Abstract

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

Author information

Contact details are reproduced from the original publication and may be historical.

Florian Schweiger

(1) Weizmann Institute of Science, Rechovot, Israel
(2) Section de Mathématiques, Université de Genève, Switzerland

florian.schweiger@unige.ch

Keywords

  • p-Capacity
  • variational problem
  • first passage percolation

Mathematics Subject Classification

31C45; 31C20, 94C15, 60K35

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Suggested citation

C. Cosco, S. Nakajima, F. Schweiger. “Asymptotics of the p-Capacity in the Critical Regime.” Journal of Convex Analysis 33 (2026), No. 1&2, 13–27. https://doi.org/10.68381/jca33002

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Published by Heldermann Verlag, 2026. Rights now held by Banach Press LLC.