Neighborhood growth dynamics on the Hamming plane

TL;DR

This paper explores neighborhood growth dynamics on the Hamming plane, analyzing minimal initial sets for full occupation and energy-entropy scaling related to initial configurations.

Contribution

It introduces a new framework for neighborhood growth on Hamming graphs and studies extremal quantities related to initial configurations and occupation probabilities.

Findings

Existence of scaling for the energy-entropy functional

Characterization of minimal sets for full occupation

Analysis of growth dynamics for large Young diagrams

Abstract

We initiate the study of general neighborhood growth dynamics on two dimensional Hamming graphs. The decision to add a point is made by counting the currently occupied points on the horizontal and the vertical line through it, and checking whether the pair of counts lies outside a fixed Young diagram. We focus on two related extremal quantities. The first is the size of the smallest set that eventually occupies the entire plane. The second is the minimum of an energy-entropy functional that comes from the scaling of the probability of eventual full occupation versus the density of the initial product measure within a rectangle. We demonstrate the existence of this scaling and study these quantities for large Young diagrams.