Identifying 'Island-Mainland' phase transition using the Euler number
  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. 
Tajkera Khatun, Tapati Dutta, Sujata Tarafdar
cond-mat.stat-mech