Random graphs and networks
The world-wide web is just one modern example of the enormous networks that we encounter in our everyday lives. Probabilistic models help us to understand how the fascinating properties of these networks emerge.
Interacting particle systems
Who will you vote for? Your decision is probably affected by the opinions of the people around you. In a similar way, many natural phenomena involve systems of particles that interact in complex ways. Mathematical models of interacting particle systems often arise from physics, biology and chemistry.
Random walks and Brownian motion
An ant in a labyrinth or a particle of pollen in a jar of water: random walks and Brownian motion have been at the very centre of probability theory for many years. Besides being interesting processes in their own right, they are also fundamental to the study of many of the other topics on this page.
Understanding complex stochastic systems often requires us to study equations where the randomness and the evolution of the system are closely linked, to the extent that new mathematical techniques are required to make sense of the solutions. Such systems arise in areas as diverse as mathematical physics and the study of financial markets, and the development of new mathematical tools are often linked to breakthroughs in the understanding of these underlying systems.
Used to model family trees, computer algorithms and the spread of disease, branching processes are also a key tool in studying many other areas of probability. In Prob-L@B we are working with industry to use branching models to improve the safety of nuclear power stations.
Percolation and stochastic geometry
Imagine water seeping through the rock beneath your feet, or coffee through a filter: what decides whether the liquid will be able to pass through the material? In a similar vein, what are the connectivity properties of random geometric graphs such as the one at the top of this page? Percolation theory and stochastic geometry address these types of question.
Counting and understanding how many structures exist with a particular collection of properties is one of the most creative areas of mathematics, and has many applications, especially to computer science. Probabilistic combinatorics includes the study of random trees, networks and Pólya urns. There are many links with the study of graphs and networks.
Many phenomena in nature, such as earthquakes or avalanches, show mathematical order - known as self-similarity - despite their chaotic nature. Self-organized criticality explains why this remarkable incongruity is, in many cases, inevitable.
Antal Járai is an expert on Abelian sandpiles and related models showing self-organized criticality, which he studies mainly through connections with uniform spanning trees and loop-erased random walk.
Understanding financial markets, including how to price assets and hedge risk, is important for everyone from supermarkets to government officials. Questions that arise from these applications are regularly difficult and lead to interesting mathematics.
Path-discontinuous stochastic processes, such as Lévy processes and self-similar Markov processes, describe microscopic movements that may jump suddenly from one location to another.
Andreas Kyprianou is one of the foremost experts on such processes, often working together with long-term visitors Juan-Carlos Pardo Millán and Victor Rivero. He is the author of the book "Fluctuations of Lévy Processes with Applications".
Monte Carlo simulation
Monte Carlo algorithms are designed to solve deterministic problems using randomness. They are used everywhere, from nuclear physics to aerospace engineering. The Department of Mathematical Sciences is also home to a large numerical analysis group, and there is strong interaction between the two groups.