Critical Exponents in Sandpiles

: (Alternative Format Thesis)

Abstract

This thesis has both numerical and theoretical aspects in studying Abelian sandpiles. We start with a numerical approach, using exact sampling. Our methods use a combination of Wilson’s algorithm to generate uniformly distributed spanning trees, and Majumdar and Dhar’s bijection. We study the probability of topplings of individual vertices in avalanches starting at the centre of large cubic lattices in 2, 3 and 5 dimensions. Based on these, we estimate the values of the toppling probability exponent in the infinite volume limit in d = 2, 3, and find good agreement with theoretical results on the mean-field value of the exponent in d ≥ 5. We also study the distribution of the number of waves in 2 dimension. Our simulation method, combined with a variance reduction concept, is well suited for analyzing various problems, even in very high dimensions. We demonstrate this by estimating the single-site height probability distribution in 32 dimension, and compare it to the asymptotic behaviour as d → ∞. Then we prove an asymptotic formula for the single-site height distribution with error estimates in terms of Poisson(1) probabilities.
We continue with studying the following problem arising in the simulation context. We consider a simple random walk on Z^d started at the origin and stopped on its first exit time from (−L, L)^d ∩ Z^d. Write L in the form L = mN with m = m(N) and N → ∞ so that L² ∼ AN^d for some real positive constant A. Our main result is that for d ≥ 3, the projection of the stopped trajectory to the N-torus locally converges, away from the origin, to an interlacement process at level Adσ₁, where σ₁ is the exit time of a Brownian motion from the unit cube (−1, 1)^d that is independent of the interlacement process. The above problem is a variation on results of Windisch (2008) and Sznitman (2009).

Date of Award	27 Apr 2022
Original language	English
Awarding Institution	University of Bath
Sponsors	ESF and EPSRC
Supervisor	Antal Jarai (Supervisor)