Scaling limits of height restricted random walks and MultiMaps
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Titel
PhD
Institution
University of Paris-Saclay
Beløb
DKK 700,000
År
2022
Bevillingstype
Internationalisation Fellowships
Hvad?
Imagine putting together a large amount of your favorite polygon together in all sorts of different ways you can. Depending on your favorite polygon, you might end up with all kinds of discrete objects: maybe trees, triangulations, quadrangulations... Now put an infinite number of them together and look around the first piece you started your construction with. That is what local limits tell you: how your object looks locally, most probably. Now take some distance and let your polygons get smaller and smaller, to the point that they would almost disappear. Your construction will be a finite object, probably a strange one. That is what scaling limits tell you: how your object looks globally, most probably. In this project, we will be investigating local and scaling limits of different classes of such random objects where some of them are trees, tall and short, wide and narrow, and some of them are maps that can admit multiple components.
Hvorfor?
Local and scaling limits of many different classes of such random objects are known, and in many cases, they happen to be similar to each other, in other words, they belong to the same universality class. Hence finding new limits is important but rather rare, and can open the way to new different interesting objects. During my PhD, we constructed such an interesting local limit for a specific class of height coupled trees in collaboration with Bergfinnur Durhuus. The aim of the project in hand is to discover new interesting limits, as they are of interest in their own right within probability theory as well as in theoretical physics, especially in quantum gravity. Apart from the theoretical framework, random trees and maps admit many applications in different fields from computer science to the study of neural networks and models of disease spreading
Hvordan?
We use pen and paper, blackboard and chalk to prove theorems. We also do simulations to see how our objects look. It also happens that one needs to develop new tools, such as mappings from a not-so-well-known family of objects to a well-known one. Hence our project requires individual effort as well as collaboration with people who have different perspectives and possibly different expertise. The probability group in University Paris-Saclay consists of leading experts working on different aspects of random geometries, which makes it an ideal place to produce diverse, high level research.