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Peter Pal Varju

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#### Dr Peter Pal Varju

## Organisation

## Research summary

## Grants awarded

#### Random walks on groups

#### Random walks on groups

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Research Fellow

University of Cambridge

My field of research is random walks on groups. A random walk is a random sequence of positions, where each position is selected randomly among the neighbours of the previous site. The systems I study come from algebra. A group is a collection of elements in which an operation similar to addition or multiplication is defined.

To describe a random walk on a group, first we choose some elements of the group that we call generators. Then we pick a random element among the generators and multiply with it the starting position. This is the first step of the random walk. Then we pick another random element of the generators multiply with it the first step. We repeat this indefinitely and obtain a sequence, which is called the random walk.

The goal of my research is to understand how the probability distributions of the steps change as the random walk evolves. A particular question that I ask is the number of steps I need to take so that I am no longer able to guess what the starting position was. This quantity is called the mixing time of the random walk.

Random walks on groups are generalizations of the theory of sums of independent random variables. The latter is a classical topic in probability theory, which is also very important in applications in many branches of science. However, the behaviour of the random walk also reflects the algebraic structure of the underlying group, and sometimes it also carries interesting number theoretic information. I believe that random walks on groups are interesting on their own right, but my research is also motivated by applications in number theory and group theory.

**Scheme:** University Research Fellowship

**Dates:** Oct 2015 - Sep 2020

**Value:** £394,006.43

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