Computer Simulations of Dimer Models

This project is the major project from my 3rd year at university.

Dimers are a representation of some entity that bonds to two sites on a lattice. Sort of like placing dominos on a chess board. By generating random configurations of these systems, then studying the statistics, you can see that gaps in the lattice (monomers) interact like charged particles!

There's a huge amount of statistical analysis in this report. And, a large amount of focus goes into methods that can reliably generate a non biased 'random' configuration. If you can stomach that, then you might like it.

Abstract

We study the statistics of dimer systems and problems with calculating expectation values. A Markov Chain Monte Carlo method that uses plaquette flipping to generate configurations is described. We observe and correct errors in our algorithm using a blocking technique, and numerical approximations of system properties are calculated with the results $\langle n_h \rangle = 0.497\pm0.005$ and $\text{var}(n_H)=0.157\pm0.005$. Motivated by a need for efficiency, a more complex loop algorithm is developed. Using this algorithm equivalent simulations are executed with improved errors, yielding $\langle n_H\rangle = 0.5004 \pm 0.0007$ and $\text{var}(n_H) = 0.1592 \pm 0.0007$. The space of the system is discussed, and we find that boundaries of the system influence numerical properties calculated in a simulation. We analyse the two-monomer system and find they behave like charged particles due to the emergence of a coulombic potential in both two and three dimensions.