A Random Walk in Statistical Physics

Doktorsavhandling, 2001

This thesis deals with some aspects of the physics of disordered systems. It consists of four papers and an introductory part. An introduction, suitable for physicists, to theoretical computer science and computational complexity is contained in chapter 2. The definition of the Turing model of computation and of some important complexity classes are given, the Church-Turing hypothesis described, and the proofs of some important theorems reviewed. Additional work by physicists on optimisation problems is described in chapter 3, while chapter 4 is an introduction to computer simulation of physical models. This chapter also contains some results for the distribution of sub-structures of lattice polymers. Chapter 5 introduces the concept of small world graphs and reviews previous work on physical models placed on such graphs. Paper I studies the relaxation of some optimisation problems that, unless a very plausible conjecture in computer science is false, have worst case instances that require exponential time to solve. These problems can also be interpreted as spin glass models, and have previously been found to exhibit threshold phenomena akin to those of physical models undergoing phase transitions. The graph colouring problem is revisited in paper IV, wherein the phase transition is studied using small world graphs. In paper II the ferromagnetic Ising model on random graphs is found to display a freezing phenomenon for a range of connectivities. Damage spreading is an important tool for studying the stability of models. Paper III finds a very good data collapse for the damage plotted as a function of temperature for small world graphs with different rewiring probabilities. Small worlds have so far almost only been studied using a 1D chain as starting lattice. The work presented in this thesis contains the first application of 2D and 3D small worlds to physical systems.