Variational principle for Gibbs point processes with finite range interaction

TL;DR

This paper proves a variational principle for Gibbs point processes with finite range interactions, showing they minimize a combined entropy and energy functional, applicable to various complex interactions.

Contribution

It establishes a general variational principle for Gibbs point processes with broad interaction types, including multibody and geometric interactions, under finite range assumptions.

Findings

Gibbs point processes minimize free excess energy

Applicable to superstable, multibody, and geometric interactions

Valid under finite range interaction assumption

Abstract

The variational principle for Gibbs point processes with general finite range interaction is proved. Namely, the Gibbs point processes are identified as the minimizers of the free excess energy equals to the sum of the specific entropy and the mean energy. The interaction is very general and includes superstable pairwise potential, finite or infinite multibody potential, geometrical interaction, hardcore interaction. The only restrictive assumption involves the finite range property.