Interacting stochastic systems

The Ising model was suggested by Lenz in 1920. It is a probabilistic model for ferromagnetism. Magnetization can then be explored as correlation between spin random variables on a graph. Bond percolation was introduced by Simon Broadbent and John Hammersley in 1957. It is a model for long range order. Edges of a lattice graph are declared open, independently, with some probability p, and clusters of open edges are studied. Both these models can be understood as aspects of the random-cluster model. In this thesis we study various aspects of mathematical statistical mechanics. In Chapter 2 we create a diluted version of the random-cluster model. This allows the coupling of the Ising model to the random-cluster model to be extended to include the Blume-Capel model. Crucially, it retains some of the key properties of its parent model. This enables much of the random-cluster technology to be carried forward. The key issue for bond percolation concerns the fraction of open edges required in order to have long range connectivity. Harry Kesten proved that this fraction is precisely one half for the square planar lattice. Recent development in the theory of influence and sharp thresholds allowed Béla Bollobás and Oliver Riordan to simplify parts of his proof. In Chapter 3 we extend an influence result to apply to monotonic measures. This allows sharp thresholds to be shown for certain families of stochastically increasing monotonic distributions, including random-cluster measures. In Chapter 4 we study time to convergence for a mean-field zero-range process. The problem was motivated by the canonical ensemble model of energy levels used in the derivation of Maxwell-Boltzmann distributions. By considering the entropy of the system, we show that the empirical distribution rapidly converges – in the sense of Kullback-Leibler divergence – to a geometric distribution. The proof utilizes arguments of spectral gap and log Sobolev type.EThOS - Electronic Theses Online ServiceGBUnited Kingdo