Continuum Percolation for Quermass Model

TL;DR

This paper studies continuum percolation in Markov germ-grain models with circular grains and Quermass interactions, demonstrating percolation occurs under broad conditions and exploring phase transitions in multi-type models.

Contribution

It introduces analysis of percolation in Quermass models with circular grains and establishes conditions for percolation and phase transition behavior.

Findings

Percolation occurs for any linear combination of Minkowski functionals with sufficiently large activity.

Percolation is guaranteed for large enough activity parameters.

Application to phase transition in multi-type Quermass models.

Abstract

The continuum percolation for Markov (or Gibbs) germ-grain models is investigated. The grains are assumed circular with random radii on a compact support. The morphological interaction is the so-called Quermass interaction defined by a linear combination of the classical Minkowski functionals (area, perimeter and Euler-Poincar\'e characteristic). We show that the percolation occurs for any coefficient of this linear combination and for a large enough activity parameter. An application to the phase transition of the multi-type Quermass model is given.