Connections in spatial growth models

The study of stochastic models for spatial growth saw its dawn in the 1960s, and led to a rigorous theory for subadditive ergodic processes.

In the 1980s, far-reaching predictions regarding the asymptotic behaviour of a large class of two-dimensional growth models that stem from the seminal work of Kardar, Parisi and Zhang further popularised the area of research.

For a handful so-called `integrable’ models, these predictions have been verified rigorously through serendipitous connections to other fields.

For the overwhelming majority of non-integrable models, understanding the mechanism that dictates the order of fluctuations remains one of the most important open problems.The current programme aims to develop general techniques through which the asymptotic behaviour of non-integrable models of spatial growth can be understood.

First-passage percolation is, perhaps, the most well-known model of this kind, and has an elegant formulation as a random metric space.

This programme will explore the central role of geodesics, i.e. distance-minimising connections, in understanding various properties of this metric space.

The main objective will be to establish rigorous connections between the asymptotic shape and fluctuations around the asymptotic shape on one hand, and geodesics on the other.

These connections will offer a novel perspective on some of the oldest and most important open problems in first-passage percolation and related models for spatial growth.