Perla Sousi, Professor of Probability in the Statistics Laboratory at the Department of Pure Mathematics and Mathematical Statistics (DPMMS), has been awarded the prestigious Whitehead Prize by the London Mathematical Society.
The Whitehead Prize is awarded annually to a number of mathematicians in the UK who are at an early stage of their career. Sousi's achievement follows a long line of Cambridge mathematicians to have been awarded the Prize. Examples include Julian Sahasrabudhe (2024) and Maria Bruna, Holly Krieger and Henry Wilton (all three in 2020).
Perla Sousi has been awarded a 2025 Whitehead Prize for 'her groundbreaking contributions to the study of mixing and cutoff phenomena for Markov chains; to the study of random walks and Brownian motion in fixed and changing environments; and to the development of the potential theory of branching random walks on d-dimensional lattices'.
Random walks and random chains
Sousi's work concerns random processes and structures which resemble phenomena we observe in real life, but come with their own mathematical interest. A dust particle floating around a room performs a random dance as it is buffeted about by collisions with other particles and molecules. Such a dance is known as Brownian motion after the botanist Robert Brown, who first observed it in pollen grains floating in water in 1827.
While entirely unpredictable, it's possible to describe Brownian motion mathematically. A so-called random walk is a sequence of discrete steps each taken in a random direction in space. There are different ways of mathematically defining the details of such a random walk, but many of them have something in common: if you let the number of steps tend to infinity while letting their size tend to zero (so that the path still fits into a finite region of space) then you end up with something called the Wiener process, named after the American mathematician Norbert Wiener. This process provides a good mathematical model for Brownian motion.
You could also perform a random walk on a network, such as the London Underground network. Each time you get to a station you decide at random which adjacent station to travel to next. This process gives an example of a so-called Markov chain, named after the mathematician Andrey Markov. A Markov chain is a sequence of events, each chosen at random, where the probability of the next event in the chain depends on the current event (in our Underground example only stations that are adjacent to the previous station can be travelled to). Markov chains find many uses inside and outside mathematics - for example they are important in Large Language Models.
Taming randomness
While it's impossible to predict what an individual random walk or Markov chain will look like, it is still possible to make general statements about them. We're familiar with this idea from real life. The law of large numbers says that, even though we can't predict the outcome of a single coin flip, we can be quite sure that the proportion of Heads and Tails in a long sequence of coin flips is roughly 50:50. The result serves to tame randomness when you're dealing with a large number of the same random event, as you might do in a casino.
In addition, a fundamental idea in probability called the central limit theorem implies that you can say something about a very large set of things (people, animals, all the shoes produced in a shoe factory) by looking at a smaller, randomly chosen sample. This is what powers opinion polls.
In her work Sousi has come up with general results, both about random walks and about Markov chains (among other things). For example, together with Amine Asselah and Bruno Schapira she derived versions of a law of large numbers and central limit theorems for types of random walks, and proved results about the so-called Wiener sausage — that's the set of points within a given distance of a Brownian path. She also showed how to characterise Markov chains that exhibit a certain fascinating property called cutoff.
The official citation notes that 'Sousi’s work on Markov chain mixing and cutoff helped to revitalize this field of study', and highlights her leading role in the development of potential theories on types of random branching walks. The prize recognises her outstanding work and its promise for the future, with the citation noting that her 'insights … will have a lasting impact on the field'. You can read the full citation here.
Congratulations to Perla Sousi on her Whitehead Prize!