Critical exponents in sandpiles

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如何引发一个长时间的活动，同时还有许多其他粒子在移动。这项研究将涉及概率论的各个领域，这些领域是当前研究的前沿：均匀生成森林，环擦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）关于d >= 5的循环擦除随机游走，Bhupatiraju，Hanson和Jarai（2016）关于波的大小的估计，以及Sznitman（2012）关于交错的一些结果。此外，在第一学期，我将参加一个阅读课程随机矩阵提供的部门和教学课程中心（TCC）课程黎曼假设。最终目标/目的和方法：目的是量化的概率大雪崩在所谓的临界指数。在我2016年夏季的本科实习中，我们在数值上找到了一个这样的临界指数的近似值，并从伦敦数学学会获得了资金，以便在2017年夏季继续这项工作。博士研究的目标是对模拟结果进行严格的数学分析。特别是，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。除了提供严格的支持使用的算法，这个问题是有趣的，在其本身的权利，因为我们希望它将揭示新的光在沙堆中的临界指数的平均场值。第二个问题是要证明严格的上限的值的指数我估计。这是一个更具挑战性的开放性问题，目前只知道下限。雪崩可以分解成所谓的波。第一部分将说明半径为L的盒子中的波不可能比L的4次方大得多。这可以通过调整现有研究的方法来实现。第二部分将利用本文的研究成果对涉及哈希的算法进行分析。其主要思想是表明，对于小波，哈希表中的探索区域近似为交错。这是具有挑战性的原因是，步行开始彼此接近。研究结果将表明，一旦随机游走远离其起点，它就会忘记它从哪里开始，并且表现得就像刚刚添加了一条独立的随机游走路径一样。最后，通过利用随机游动及其相交概率的工具，这将扩展到更大的波。参考文献：Bhupatiraju，S.，汉森，J.，和加赖，AA，2016. D维沙堆中临界指数的不等式[在线]。可从：https://arxiv.org/abs/1602.06475v1 [2016年2月20日访问]Lawler，G.F.，1991.随机漫步的交叉点纽约：施普林格科学+商业媒体。2012.随机交错和高斯自由场。《概率年鉴》，40，2400-2438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.