Critical exponents in sandpiles

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.