Discrete Quantitative Isocapacitary Inequality: Fluctuation Estimates

TL;DR

This paper establishes quantitative fluctuation estimates for the discrete isocapacitary problem on subsets of integer lattices, extending continuum inequalities to the discrete setting as set size grows.

Contribution

It introduces a method to extend continuum isocapacitary inequalities to the discrete lattice setting, providing fluctuation estimates for large sets.

Findings

Quantitative fluctuation estimates for discrete isocapacitary problem

Extension of variational problem from discrete to continuum

Sharp continuum inequalities applied to discrete sets

Abstract

The classical isocapacitary inequality states that, among all sets of fixed volume, the ball uniquely minimizes the capacity. While this result holds in the continuum, it fails in the discrete setting, where the isocapacitary problem may admit multiple minimizers. In this paper we establish quantitative fluctuation estimates for the discrete isocapacitary problem on subsets of $\mathbb{Z}^d$ as their cardinality diverges. Our approach relies on a careful extension of the associated variational problem from the discrete to the continuum setting, combined with sharp (continuum) quantitative isocapacitary inequalities.