The Expected Perimeter in Eden and Related Growth Processes

TL;DR

This paper investigates the expected perimeter in Eden growth processes and related models, providing bounds and contrasting behaviors through theoretical analysis and previous results in percolation and statistical mechanics.

Contribution

It establishes an upper bound on the expected perimeter in Eden growth processes and demonstrates that certain lattice animal growth histories do not follow this bound.

Findings

Upper bound on expected perimeter in Eden growth processes

Contrasting perimeter behavior in different growth histories

Insights into lattice animal growth in statistical mechanics

Abstract

Following Richardson and using results of Kesten on First-passage percolation, we obtain an upper bound on the expected perimeter in an Eden Growth Process. Using results of the author from a problem in Statistical Mechanics, we show that the average perimeter of the lattice animals resulting from a very natural family of "growth histories" does not obey a similar bound.