Work

Asymptotics of the p-Capacity in the Critical Regime

Year: 2026

Type: article

Abstract: We are interested in the asymptotics of the p p -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.

Source: Journal of convex analysis

DOI: 10.68381/jca33002

Authors: Clément Cosco, Shuta Nakajima, Florian Schweiger

Institutions: Université Paris Dauphine-PSL, Meiji University, University of Geneva, Weizmann Institute of Science

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: bronze

Funders: European Commission, Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung, Israel Academy of Sciences and Humanities

Award: Hierarchical structure and the extrema of random fields