Derivative formula for capacities

TL;DR

This work derives derivative-type asymptotics for several notions of capacity on the lattice when two finite sets are placed far apart. It treats Newtonian capacity with an exact cross-term expansion, Cap_α via variational representations, and branching capacity through spine decompositions of branching random walks and harmonic/escape-measure connections. The main results establish that the deficit in capacity from the union scales as twice the product of the individual capacities, with the appropriate kernel (g(z) or G(z)) governing the asymptotics. These findings illuminate how capacities interact at large distance and may impact high-dimensional percolation and related processes.

Abstract

We obtain a derivative formula for various notions of capacity. Namely we identify the second order term in the asymptotic expansion of the capacity of a union of two sets, as their distance goes to infinity. Our result applies to the usual Newtonian capacity in the setting of random walks on the Euclidean lattice, to the family of Bessel-Riesz capacities, and to the Branching capacity, which has been introduced recently by Zhu [9] in connection with critical Branching random walks. On the other hand, the result remains open for the notion of capacity in the setting of percolation, which is introduced in a companion paper, but serves as a motivation, as it would have some interesting consequences there.

Theorems & Definitions (14)

• Definition 4.1
• Remark 4.2
• Claim 4.3
• proof : Proof
• Lemma 4.4
• Remark 4.5
• proof : Proof of Lemma \ref{['lem:hitfinitetoinfinite']}
• Lemma 4.6
• Lemma 4.7
• Lemma 4.8