Identifying 'Island-Mainland' phase transition using the Euler number

TL;DR

This paper investigates the 'Island-Mainland' phase transition in a lattice system, demonstrating that the Euler number's minimum indicates the transition point, which occurs after the percolation threshold and relates to physical phenomena.

Contribution

It introduces the use of the Euler number to identify the Island-Mainland transition in lattice systems, extending understanding beyond traditional percolation analysis.

Findings

Euler number reaches a minimum at the transition point

Transition occurs at a concentration higher than percolation threshold

The phenomenon relates to physical systems' experimental observations

Abstract

In the present communication we describe the Island-Mainland transition, occurring in a square lattice, when black squares are randomly dropped on a white background. Initially clusters of black squares are observed on the connected white background. But as concentration of black sites increases, at some point the black squares join to form a single continuous black background with white clusters randomly scattered in it. We show that the minimum in the Euler number, defined as the difference between number of white clusters and number of black clusters, reaches a minimum at this point. This occurs at a concentration higher than the well-known percolation phase transition and we show that the phenomenon can be related to experimental observations in several physical systems.