Critical exponents in sandpiles

一般领域：阿贝尔沙堆是雪崩的数学模型，其中粒子根据简单的规则移动。最基本的挑战是理解一个新粒子的加入是如何引发大量其他粒子运动的长时间活动的。这项研究将涉及当前研究前沿的概率论的各个领域：均匀跨越森林，循环删除随机行走和随机交错。早期训练（6 - 12个月）：我需要学习Lawler（1991）在d b> = 5中的循环擦除随机漫步，Bhupatiraju， Hanson和Jarai（2016）对波浪大小的估计，以及Sznitman（2012）对交错的一些结果。此外，在第一学期，我将参加由部门提供的关于随机矩阵的阅读课程和教学课程中心（TCC）的课程Riemann’s Hypothesis。最终目标/目的和方法：目的是根据所谓的临界指数来量化大型雪崩的概率。在我2016年夏天的本科实习中，我们从数字上找到了一个这样的关键指数的近似值，并从伦敦数学学会获得了资金，在2017年夏天继续这项工作。博士研究的目标是对模拟结果进行严格的数学分析。特别地，将考虑与模拟有关的两个方面。第一个是对今年夏天设计和使用的算法进行严格分析，并证明其平均运行时间的上限。除了为该算法的使用提供严格的支持外，这个问题本身也很有趣，因为我们期望它将为沙堆中关键指数的平均场值提供新的线索。第二个问题是证明我估计的指数值的严格上界。这是一个更具挑战性的开放问题，目前只知道它的下界。最初，将有三个部分来开展这项研究。雪崩可以分解成所谓的波浪。第一部分将证明半径为L的方框中的波不可能比L的4次方大很多。这可以通过调整现有研究的方法来实现。第二部分将使用这项研究来分析涉及哈希的算法。其主要思想是表明，对于小波，哈希表中的探索区域大约是一个交错。这之所以具有挑战性，是因为步行的起点彼此很近。研究结果将表明，一旦随机行走远离其起点，它就会忘记它的起点，并表现得好像刚刚添加了一个独立的随机行走路径。最后，这将扩展到更大的波通过绘制工具从随机游走和他们的交集概率。参考文献：Bhupatiraju， S., Hanson， J., and Jarai, a.a., 2016。d维沙堆中临界指数的不等式[在线]。可从：https://arxiv.org/abs/1602.06475v1[访问日期：2016年2月20日]Lawler， g.f., 1991。随机游走的交叉点。纽约：b施普林格科学+商业媒体。Sznitman, a.s., 2012。随机交错与高斯自由场。《概率论年鉴》，40，第2400-2438页。

General field:The Abelian sandpile is a mathematical model for avalanches, in which particles move around according to simple rules. The basic challenge is to understand how the addition of a new particle triggers a long period of activity with lots of other particles moving. The study will involve various areas of probability theory that are at the forefront of current research: uniform spanning forests, loop-erased random walks and random interlacements. Early training (6 - 12 months):I will need to learn about the loop-erased random walk in d >= 5 by Lawler (1991), the estimates on the size of waves by Bhupatiraju, Hanson and Jarai (2016), and some results on interlacements by Sznitman (2012). In addition, in semester 1, I will attend a reading course on Random matrices offered by the department and the taught course centre (TCC) course Riemann's Hypothesis.Final goals/aims and methodology:The aim is to quantify the probability of large avalanches in terms of so-called critical exponents. In my undergraduate internship in Summer 2016, we found numerically an approximate value of one such critical exponent, and the funding was received from the London Mathematical Society to continue the work in Summer 2017. The goal of the PhD research is to provide a rigorous mathematical analysis of the results from the simulations. In particular, two aspects connected to the simulation will be considered. The first one is a rigorous analysis of the algorithm designed and used this summer, and to prove an upper bound on its average running time. In addition to lending rigorous support to the use of the algorithm, this problem is interesting in its own right, since we expect that it will shed new light on the mean-field values of critical exponents in sandpiles. The second problem is to prove rigorous upper bounds for the values of the exponent I estimated. This is a much more challenging open question for which only lower bounds are known at the moment.Initially, there will be three parts to carry out this research. The avalanche can be decomposed into so-called waves. The first part will be to show that waves in the box of radius L cannot be much bigger than L to the power of 4. This can be done by adapting methods from existing research. The second part will be to use this research to analyse the algorithm which involved hashing. The main idea is to show that, for small waves, the explored region in the hash table is approximately an interlacement. The reason why this is challenging is that the walks are started near each other. The research results will show that once a random walk is away from its starting point, it forgets where it started and behaves as if an independent random walk path has just been added. Finally, this will be extended to larger waves by drawing on tools from random walks and their intersection probabilities. References:Bhupatiraju, S., Hanson, J., and Jarai, A.A., 2016. Inequalities for Critical Exponents in d-dimensional Sandpiles [Online]. Available from: https://arxiv.org/abs/1602.06475v1 [Accessed 20 Feb 2016]Lawler, G.F., 1991. Intersections of Random Walks. New York: Springer Science+Business Media.Sznitman, A.S., 2012. Random Interlacements and the Gaussian Free Field. The Annals of Probability, 40, pp 2400-2438.