We present a random interface model on the one-dimensional torus of size $N$ with a weak perturbation, i.e. an asymmetry $\sim N^{-\gamma}$ of the direction of growth that switches from up to down based on the sign of the area underneath. The evolution of the interface can be studied in terms of the density field of an underlying, non-Markovian exclusion process. We compute the order of the correlation functions of this process for the invariant measure of the interface model, and investigate the stationary fluctuations of the density field: we establish the convergence to an Ornstein-Uhlenbeck equation for $\gamma>\frac{8}{9}$, and discuss the limit for $\frac{1}{2}\leq \gamma<\frac{8}{9}$. Based on joint work with Martin Hairer and Patrícia Gonçalves.

Reaction-diffusion equations arise naturally when modelling multi-component systems of interacting populations. These equations are widely employed to describe pattern formation phenomena across various biological, chemical and physical processes. The kinetic theory of statical mechanics provides a powerful framework to describe different types of interactions at multiple spatial or temporal scales. Through appropriate hydrodynamic limits of the kinetic systems, macroscopic equations can be derived, describing observable quantities and explaining how macroscopic phenomena emerge from the underlying microscopic dynamics. In this talk, I will apply these tools to study the evolution and interactions of competing bacterial populations on a leaf surface. Specifically, I will consider self and cross diffusion effects and investigate Turing instability properties leading to the formation and persistence of stationary spatial patterns.

This work is a collaboration with D. Cusseddu (University of Minho), M. Bisi and R. Travaglini (University of Parma, Italy).