Consider a two-dimensional lattice where each lattice site is given a random weight according to some background probability distribution. Given two lattice points, the maximum over all possible paths connecting them of the sum of the weights is called last passage time and it is the purpose of last passage percolation models to study the properties of the last passage time.
Models of last passage percolation are well-studied, particularly when they exhibit certain integrability properties that make them analytically tractable. The importance arises from rigorous connections between a handful of those models and random matrix theory. Limiting laws for the last passage time (the main object of study) are the same as those for the limiting laws for the largest eigenvalue of certain random matrix ensembles. These are called Tracy-Widom laws and they are synonymous with membership of the Kardar-Parisi-Zhang (KPZ) class. While it is expected that a wide range of models belong to this class, this universality result has remained so far elusive. The proposed projects delve into a model that is partially in the KPZ class but already exhibits behaviour that it is not expected or predicted by the fully integrable models.
· The law of large numbers for the last passage time is cast as a variational formula, and when explicit calculations can be performed (depending on the discontinuities of the macroscopic speed function) there is strong suggestion that the variance of the model behaves in various different ways. In fact, heuristic numerical evidence suggests that the model can simultaneously have two directions with difference order for the variance. The idea is to rigorously understand this behaviour, in as general a way as we can.
· There are several examples in which the limiting last passage constant (the scaling limit of the last passage time up to point (nx, ny)) exhibits flat regions. Flat regions in last passage percolation arise very rarely when the environment has continuous distributions, and they affect the shape of the large deviation rate function (a measure of how rare an exponentially unlikely event is). As a first approach, we would like to compute the rate function explicitly for some models, and then show its existence and properties. Moreover, we can ask the same questions for the maximal paths and shape of maximisers for the variational formula. This aspect of the project will also involve a serious application of Monte Carlo simulations.
For more information, please see:
1) Federico Ciech and Nicos Georgiou. Last passage percolation in an exponential environment with discontinuous rates. https://arxiv.org/abs/1808.00917
2) Nicos Georgiou, Rohini Kumar, and Timo Seppäläinen: TASEP with discontinuous jump rates.
ALEA 7: 293-318, 2010. https://arxiv.org/abs/1003.3218
Students are encouraged to apply, or at least send an expression of interest, as early as possible. The studentship will be filled as soon as a suitable candidate can be found.
Possible start dates are 15 September 2019, 15 January 2020, or 15 May 2020.
The last possible date that applications can be considered is 31 August 2019.
Informal enquiries are welcome and should be directed to Dr. Nicos Georgiou [email protected]
(Profile page: http://www.sussex.ac.uk/profiles/329373
) and Prof. Enrico Scalas, [email protected]
(Profile page: http://www.sussex.ac.uk/profiles/330303
How to Apply
Apply through the Sussex on-line postgraduate application system accessible from: https://www.sussex.ac.uk/study/phd/degrees/mathematics-phd https://www.sussex.ac.uk/study/phd/apply
In the Other Information/Funding Section state that you are applying for "PhD studentship on Large Deviation Rate Function and Variance for Inhomogeneous Corner Growth Processes with Dr Nicos Georgiou and Prof Enrico Scalas".
Finally, send an email to Dr Georgiou ([email protected]
) and Prof Scalas ([email protected]
) to confirm that you have applied.