The topology of interfaces

Abstract

A discrete model of an interface separating two phases is considered. The interface is defined as the boundary of a self-avoiding n-omino with volume n embedded in the hypercubic lattice. The author proves that if there is no surface tension in the interface, then it has 'non-trivial' topology with probability one in the scaling limit. Here non-trivial topology means that the interface will consist of several disjoint components and will have non-zero genus. Moreover, the total area of the n-omino will be proportional to its volume.