The role of noise in optimisation and diffusion limited aggregation
This thesis focuses on the role played by fluctuations in both thermal optimisation techniques and diffusion limited aggregation. The key idea is that by tuning the level of input noise asymptotic results may be attained more rapidly. Stochastic optimisation problems are considered, where the function being optimised cannot be known exactly and may only be estimated. The noise in the estimates is used as the analogue of thermal fluctuations in simulated annealing. This analogy is made exact by use of an acceptance function, and stochastic annealing is seen to be the generalisation of simulated annealing to stochastic optimisation problems. The probabilistic travelling salesman problem (PTSP) is used as a test-bed for stochastic annealing, and significant new results are found. A good characterisation is found for the PTSP and scaling arguments are shown to be accurate for determining the expected length of the pruned and a priori tours, specifically E(Lpruned)=βpruned(p)√np. An oil field project, as a complex commercial problem, is considered and stochastic annealing is seen to make a large improvement in the expected return of the project. Noise reduction in diffusion limited aggregation (DLA) is known to be crucial to our understanding. A generalisation of noise reduction off-lattice is introduced, and noise reduction is shown to be a central parameter controlling the growth of DLA. In 2 dimensions, all quantities appear to be influenced by the slowest correction to scaling, with exponent ~ 0.3. In 3 dimensions, some quantities are not affected by the slowest correction to scaling, exponent ~ 0.2. The renormalisation of DLA is considered, and the noise reduction at the fixed point is measured. The noise, given as the relative variability in extremal cluster radius is found to be ε*2D ≃ 0.003 and ε*3D ≃ 0.006 in 2 and 3 dimensions, respectively.