Enumeration Methods and Series Analysis of Self-Avoiding Polygons on the Hexagonal Lattice, with Applications to Self-organizing Particle Systems

We consider programmable matter as a collection of simple computational elements (or particles) that self-organize to solve system-wide problems of movement, configuration, and coordination. Here, we focus on the compression problem, in which the particle system gathers as tightly together as possible, as in a sphere or its equivalent in the presence of some underlying geometry. Within this model a configuration of particles can be represented as a unique closed self-avoiding walk on the triangular lattice. In this paper we will examine the bias parameter of a Markov chain based algorithm that solves the compression problem under the geometric amoebot model, for particle systems that begin in a connected configuration with no holes. This bias parameter, $\lambda$, determines the behavior of the algorithm. It has been shown that for $\lambda > 2+\sqrt{2}$, with all but exponentially small probability, the algorithm achieves compression. Additionally the same algorithm can be used for expansion for small values of $\lambda$; in particular, for all $0 < \lambda < \sqrt{\tau}$, where $\lim_{n\to\infty} {(p_n)^{1
}}=\tau$. This research will focus on improving approximations on the lower bound of $\tau$. Toward this end we will examine algorithmic enumeration, and series analysis for self-avoiding polygons.