Critical exponents in sandpiles

Lead Research Organisation: University of Bath
Department Name: Mathematical Sciences

Abstract

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.

Publications

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publication icon
Járai A (2019) Toppling and height probabilities in sandpiles in Journal of Statistical Mechanics: Theory and Experiment

Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509589/1 30/09/2016 29/09/2021
1943826 Studentship EP/N509589/1 30/09/2017 30/03/2021 Minwei SUN
 
Description The Abelian sandpile is a mathematical model for avalanches, in which particles move around according to simple rules. The fundamental challenge is understanding how adding a new particle triggers a long period of activity with lots of other particles moving. The study involved various areas of probability theory that are at the forefront of current mathematical research. We analyzed the probability that adding a single particle causes a disturbance at a distant point. We gave the form of its dependence on the distance based on numerical simulations. We also considered the distribution of the number of particles at a single point and gave a formula for this in high dimensions. We proved a mathematical result that provides a heuristic understanding of the algorithm's running time for the high-dimensional simulations.
Exploitation Route We formulated some conjectures about the toppling probabilities, which would be interesting to investigate further. We give a heuristic upper bound for the algorithm's running time, which others can explore further.
Sectors Other

URL https://arxiv.org/abs/2108.12629
 
Title Simulation data for toppling and height probabilities in sandpiles 
Description This dataset provides simulation data used in analyzing the results in Chapter 3 simulation results in the thesis Critical Exponents in Sandpiles by Minwei Sun. This dataset contains simulation data in 2d, 3d, 5d, and 32d in folders named sandpile_data_xd. The characteristics simulated include the toppling probability, the number of waves, and the height probability at the origin. 
Type Of Material Database/Collection of data 
Year Produced 2022 
Provided To Others? Yes  
Impact The dataset is to accompany the publication: Toppling and height probabilities in sandpiles. 
URL https://researchdata.bath.ac.uk/id/eprint/1088