Mixing times of Markov chains

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My main research is in discrete probability, in particular the area of mixing times of Markov chains; these chains have nice properties if they are run for `a long time', and my research asks the question "how long is `a long time' ?". This has applications in a wide range of fields, including the following: - statistical physics, which deals with physics problems with inherent randomness, and has applications in many elds of physics, biology, chemistry, neurology and even some social sciences, such as sociology;- computer sciences (CS), in particular using randomised algorithms, rather than deterministic ones, and knowing for how long one should run such an algorithm { the fields of machine learning and artificial intelligence use such algorithms;- statistics, in particular when wanting to sample from a complicated distribution one can instead sample from an approximation to this using a `Markov chain', but one would like to have bounds on how long to run the chain so that the approximation is valid.In particular, my work and future work includes some of the following problems.-Networking problems. optical networks (for example, broad-band along breoptics). I have done some work on random routing algorithms for breoptic networks. To this end I introduced a new model, which combines one model with another to produce an extension of the latter, and am looking at properties of this model. There are many papers, particularly in CS and EE, working on the two individualmodels that I combined, but, to the best of my knowledge, the same work is not done on the extended model. My research can be used to show `stability' in the performance of such a network, in the sense that, subject to certain constraints, it is able to process all the data being asked of it without long delays.-Statistics. A statistics paper was recently released introducing a new method for sampling approximately from a given (complex) distribution. In the paper, the authors use this algorithm, but analysis of the algorithm is not the focus of the paper; rather, the authors focus on the application of the algorithm. As such, they do not have quantitative bounds on how well the algorithm approximatesthe real distribution { and indeed, there can be subtitles where the algorithm may appear to be working, but closer inspection shows that it is in fact not.Looking into the performance of this algorithm is a line of research that my supervisor and I intend to study at the start of this upcoming academic year.-More abstract probability. I have also worked on more abstract problems, with less applicability. In particular, I have just submitted a paper on `random motion of a particle on a randomly evolving network', and am in the final stages of another paper where a network is chosen randomly and a particle moves randomly around this (fixed) network.