Discrete Quantitative Isocapacitary Inequality: Fluctuation Estimates

TL;DR

The paper studies the discrete isocapacitary problem on $\mathbb Z^d$, where minimizers may fail to be rigid as in the continuum. It develops a discrete-to-continuum strategy: embed a finite lattice configuration into a continuum set via the Kuhn decomposition, compare the discrete capacity with the continuum $p$-capacity using $\Gamma$-convergence, and apply sharp continuum quantitative isocapacitary inequalities to obtain fluctuation bounds. The main results show that minimizers are close (up to translations) to discrete balls with symmetric-difference controlled by $N^{1-\frac{1}{2d}}$, and almost-minimizers obey a bound involving $\alpha_N$ and a scaled perimeter $P_N(X)$; the authors also prove uniform diameter and perimeter bounds and develop a robust discrete rearrangement framework to deduce structural properties. Together, these findings provide deterministic geometry controls for near-optimal discrete configurations and offer potential applications to probabilistic lattice models where capacity governs large-scale rare events.

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.

Figures (5)

• Figure 1: Construction of the function $u_0$ from $u$: The connected component of $\{u=\alpha_m\}$ is shifted to be adjacent to $0$ and the values in between are shifted to start where the new component $\{u_0=\alpha_m\}$ ends.
• Figure 2: The function $u_k$ obtained from $u$: $\{u_k \leq \alpha_k\} = -\{u \leq \alpha_k\}$.
• Figure 3: The rearrangement of $v$ with respect to $u$: The highest value of $v$ is rearranged in order to be paired with the highest value of $u$.
• Figure 4: The two permutations introduced in Step 1. The dashed lines indicate the interactions
• Figure 5: The construction in Step 2: The bold black line is contained in $X$, whereas the two bold gray lines are contained in $X^{*(e_k+e_{k+1})}$. Hence, also the light gray square is contained in $X^{*(e_k+e_{k+1})}$.

Theorems & Definitions (53)

• Definition 2.1: Convexity
• Remark 3.1
• Theorem 3.2
• Definition 3.3
• Proposition 3.4
• proof
• Definition 3.5
• Proposition 3.6
• proof
• Theorem 4.1