Mixing times of Markov chains

我的主要研究方向是离散概率，特别是马尔可夫链的混合时间；如果这些连锁店运行“很长时间”，它们就会有很好的特性，我的研究提出了一个问题：“多长时间是‘很长时间’？”这在许多领域都有应用，包括：-统计物理学，它处理具有内在随机性的物理问题，并在物理学，生物学，化学，神经学甚至一些社会科学，如社会学的许多领域都有应用；-计算机科学（CS），特别是使用随机算法，而不是确定性算法，并知道应该运行这种算法多长时间（机器学习和人工智能领域使用这种算法）；-统计，特别是当想要从复杂的分布中抽样时，可以使用“马尔可夫链”来代替从近似中抽样，但人们希望有多长时间运行链的界限，以便近似是有效的。特别是，我的工作和今后的工作包括以下一些问题。网络问题。光网络（例如沿breoptics的宽带）。我对光纤网络的随机路由算法做了一些研究。为此，我引入了一个新模型，它将一个模型与另一个模型结合起来，产生了后者的扩展，并且正在研究这个模型的属性。有很多论文，特别是在CS和EE方面，都在研究我所结合的两个单独的模型，但是，据我所知，在扩展模型上并没有做同样的工作。我的研究可以用来展示这种网络性能的“稳定性”，也就是说，在一定的约束下，它能够处理所有被要求的数据，而不会有很长的延迟。最近发表了一篇统计学论文，介绍了一种从给定（复杂）分布中近似抽样的新方法。在本文中，作者使用了该算法，但对算法的分析并不是本文的重点；相反，作者关注的是算法的应用。因此，对于算法接近真实分布的程度，他们并没有定量的界限，实际上，算法可能在某些地方看起来是有效的，但仔细检查就会发现它实际上不是。研究这种算法的性能是我的导师和我打算在即将到来的学年开始时研究的一个研究方向。-更抽象的概率。我也研究过一些更抽象的问题，适用性更弱。特别是，我刚刚提交了一篇关于“随机进化网络上粒子的随机运动”的论文，并且在另一篇论文的最后阶段，其中一个网络是随机选择的，一个粒子在这个（固定）网络上随机移动。

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.