Directed Percolation and the Abstract Tile Assembly Model

Abstract

Self-assembly is a process by which simple components build complex structures through local interactions. Directed percolation is a statistical physical model for describing competitive spreading processes on lattices. The author describes an algorithm which can transform a tile assembly system in the abstract Tile Assembly Model into a directed percolation problem, and then shows simulations of the aTAM which support this algorithm. The author also investigates two new constructs designed for Erik Winfree\u27s abstract Tile Assembly Model called the NULL tile and temperature 1.5. These constructs aid the translation between self-assembly and directed percolation and may assist self-assembly researchers in designing tilesets in the aTAM with non-deterministic local properties, but guaranteed global properties. Temperature 1.5 results indicate the brittleness of the standard temperature 2 tile assembly system, and the NULL tile is shown to assist simulations of large assembly processes while also reinforcing the need for variable temperature models to more closely simulate laboratory self-assembly