Complex stochastic systems.

Lead Research Organisation: Durham University
Department Name: Mathematical Sciences

Abstract

Many real-world systems exhibit intrinsic randomness at the level of the smallest components, and these components interact in complicated ways to produce macroscopic phenomena. This project will use modern probabilistic techniques to study such systems.

Publications

10 25 50

Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509462/1 01/10/2016 30/09/2021
1850137 Studentship EP/N509462/1 01/04/2017 31/12/2020 Clare Wallace
 
Description We study the behaviour of a random walk in two dimensions, which jumps between integer points. The x-coordinate must always increase; the y-coordinate has no such restriction.

Under the fairly relaxed assumption that longer jumps (in either direction) have exponentially smaller probability, we examine the behaviour of a section of the trajectory of width n. We fix two properties of the trajectory, conditioning on both its endpoint and its integral being far from their means.

As n grows, we are able to categorise the behaviour of the conditional trajectories on two scales. If our x- and y- coordinates zoom at the same scale, we see a limiting profile c(t) which we can categorise in terms of
- the distribution of the jumps
- our fixed endpoint
- our fixed integral

When our x-coordinate scales like the square root of the y-coordinate, the difference between our trajectory and its limiting profile converges to a Brownian bridge.
Exploitation Route The model, as described above, is a good approximation of the skeleton of the phase boundary in the Ising model with fixed total magnetisation. It could be used to derive sharper asymptotics for this phase boundary. Since the Ising model is a special case of many of the models currently studied in mathematical physics, these asymptotics could have a wide range of applications.
Sectors Digital/Communication/Information Technologies (including Software)