Critical exponents in sandpiles

The Abelian sandpile is a mathematical model for avalanches, in which particles move around according to simple rules. The fundamental challenge is to understand how adding 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.
The project will include both numerical and theoretical aspects in studying Abelian sandpiles. We will begin with a numerical approach to study the probability of topplings of individual vertices in avalanches starting at the centre of large cubic lattices in two and three dimensions. Based on these, we estimate the values of the toppling probability exponents in the infinite volume limit. Additionally, we will analyze various characteristics of sandpile models in two and three dimensions. We will attempt to expand our code to high-dimensional settings. Then, we will focus on theoretical problems that arise in the simulation context, such as the asymptotic behaviour of the single-site height distribution as dimension d tends to infinity. Further, we would like to conduct a rigorous analysis of the algorithm designed and implemented and provide 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.