Probabilities in a flat world

Tomas Berggren will receive his doctoral degree in mathematics from KTH Royal Institute of Technology in 2020. Thanks to a grant from Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position with Professor Alexei Borodin at the Massachusetts Institute of Technology (MIT) in Boston, USA.

Much of modern probability research seeks to understand the behavior of large and complex random systems, such as turbulent fluids, crack formation, or traffic flows.  These systems are usually far too large and complex to be fully covered by computer modeling. Instead, simpler models are created for these systems, reflecting essential aspects of all the phenomena studied, while also being simple enough to allow accurate mathematical calculations. 

A family of models that have proved particularly useful is a type of random tiling, where a flat surface is covered with rectangular domino bricks. A well-known and well-studied example is covering a square with bricks. Although the bricks are laid out randomly, it has been found that a structure can be found in all the tiling patterns, provided that the area covered is very large. 

This is true for a development of this model, in which a surface called the Aztec diamond is studied. An Aztec diamond is a rotated square with stepped edges and, if it is large enough, there will almost always be a circular structure in the domino bricks’ tiling pattern. Outside the circle, the bricks are ordered, while inside it they are disordered.

Given an area and type of tile, we can also allocate a weighting to selected tiles, so that some tiles become more probable than others. Completely new phenomena have been discovered in such weighted models. However, very little is known about which phenomena can be expected to emerge, why they do so and which of these phenomena also appear in other contexts. Tomas Berggren’s project aims to improve the understanding of these weighted models.