Mixing times of Markov chains

我的主要研究是离散概率，特别是马尔可夫链的混合时间区域；如果这些链运行一段时间，就会有很好的性质，我的研究提出了这样一个问题：“长时间是多长？”这在广泛的领域都有应用，包括：-统计物理学，它处理具有固有随机性的物理问题，并在物理、生物、化学、神经学甚至一些社会科学的许多领域中都有应用，例如社会学；-计算机科学(CS)，特别是使用随机算法，而不是确定性算法，并且知道一个人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；-统计学，特别是当想要从一个复杂的分布中抽样时，人们可以改用‘马尔科夫链’从与此近似的样本中进行抽样，但人们希望有一个界限，即运行链多长时间，以便近似有效。特别是，我的工作和未来的工作包括以下一些问题。-网络问题。光网络(例如，宽带光纤)。我在光网络的随机路由算法方面做了一些工作。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，以产生后者的扩展，并正在研究该模型的性质。有很多论文，特别是在CS和EE中，研究了我结合的两个单独的模型，但就我所知，同样的工作没有在扩展的模型上完成。我的研究可以用来证明这样一个网络性能的“稳定性”，在某种意义上，在一定的约束下，它能够处理要求它处理的所有数据，而不会有长时间的延迟。最近发表了一篇统计论文，介绍了一种从给定(复杂)分布近似抽样的新方法。在本文中，作者使用了该算法，但对该算法的分析并不是本文的重点，而是将重点放在该算法的应用上。因此，他们对算法接近真实分布的程度没有量化的界限{事实上，在字幕中算法可能看起来有效，但更仔细的检查发现它实际上并不有效。我和我的导师打算在下一学年开始时研究这个算法的性能。-更抽象的概率。我还研究过更抽象的问题，适用性较差。具体地说，我刚刚提交了一篇关于《随机演化网络上粒子的随机运动》的论文，我现在正处于另一篇论文的最后阶段，在这篇论文中，网络是随机选择的，粒子在这个(固定)网络上随机运动。

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.