Random walks

Random walks are a central pillar of probability theory, a topic of beautiful pure mathematical research but which also have a wide variety of applications, from finance to medicine. A random walk describes a process that evolves step by step, where each step is chosen at random from a specified distribution. Despite this simplicity, random walks give rise to remarkably rich and complex behaviour. One striking early result was proven by George Pólya in 1921, showing that in one and two dimensions, a simple random walk will return to its starting point infinitely often, whereas in three and more dimensions, it will eventually drift away indefinitely.

More recently, many variants of random walks have been studied, sometimes motivated by applications: loop-erased random walks, self-avoiding walks, reinforced random walks, random walks in random environments, and dynamical random walks. Techniques from the study of random walks have been useful in many contexts, and methods from other areas, for example percolation, have been used to understand random walks.

Branching structures

Branching structures are another key ingredient in many aspects of research and applications of probability theory. Imagine a family tree, which can be modelled by assuming that each person has a random number of children; will the tree survive forever, or eventually die out?

These structures are essential in the study of evolution, but also have more surprising applications to telecommunication networks and nuclear power. They also have many applications to other parts of probability and wider mathematics. However, often the applications - whether to real-life problems or some other mathematical theory - have only an approximate branching structure. It may be that the number of branches is likely to be larger in some parts of the structure than others, or that the distribution depends on the structure elsewhere. These complications lead to very technical and delicate research that has been a speciality of ProbLaB for many years.

Members of ProbLaB with research in these areas include:

• Antal Járai, who has results on the number of particles that visit a site in critical branching random walks.
• Daniel Kious, who has studied random walks in random environments, where the step distribution itself is random; particularly reinforced random walks, where the step distribution depends on the number of times the walk has visited each site.
• Cécile Mailler, who has worked on random trees that emerge from computer algorithms, random walk models of reinforcement learning, and Pólya urns.
• Marcel Ortgiese, who is interested in branching processes in random environments, where the branching structure is heavily influenced by local inhomogeneities, and in random graphs where the local picture is described by a branching process.
• Christoforos Panagiotis, who has studied self-avoiding walks where the process cannot visit sites it has already visited, as well as Bienaymé-Galton-Watson trees.
• Sarah Penington, whose work on branching structures is often motivated by population genetics, but has also provided new results and techniques in the study of partial differential equations.
• Matt Roberts, who has worked on Bienaymé-Galton-Watson trees, branching random walks and branching Brownian motion, as well as dynamical random walks.
• Florian Schweiger, who has used estimates from branching random walks to study the maxima of random fields.