We consider a class of interacting particle systems where particles perform independent random walks and spread an infection according to a susceptible-infected-recovered model. I will discuss a new method for understanding this model and some variants. A highlight of this method is that if recovery rate is low, then the infection survives forever with positive probability, and spreads outwards linearly leaving a herd immunity region in its wake. Based on joint work with Allan Sly.

We consider a natural random growth process with competition on $Z^d$ called first-passage percolation in a hostile environment, that consists of two first-passage percolation processes $FPP_1$ and $FPP_\lambda$ that compete for the occupancy of sites. Initially $FPP_1$ occupies the origin and spreads through the edges of $Z^d$ at rate $1$, while $FPP_\lambda$ is initialised at sites called seeds that are distributed according to a product of Bernoulli measures of parameter $p$. A seed remains dormant until $FPP_1$ or $FPP_\lambda$ attempts to occupy it, after which it spreads through the edges of $Z^d$ at rate $\lambda$. We will discuss the results known for this model and present a recent proof that the two types can coexist (concurrently produce an infinite cluster) on $Z^d$. We remark that, though counterintuitive, the above model is not monotone in the sense that adding a seed of $FPP_\lambda$ could favor $FPP_1$. A central contribution of our work is the development of a novel multi-scale analysis to analyze this model, which we call a multi-scale analysis with non-equilibrium feedback and which we believe could help analyze other models with non-equilibrium dynamics and lack of monotonicity. Based on a joint work with Tom Finn (Durham University).

Take a point $x$ on the 2-dimensional integer lattice and another one $y$ North-East from the first. Place i.i.d. Exponential weights on the vertices of the lattice; the last passage time between the two points is the maximal sum of these weights which can be collected by a path that takes North and East steps. The process of these weights as y varies is a difficult one, but locally has a nice asymptotic structure. I'll explain what stationary queues have to do with this and how this insight gives coalescence properties of the maximal-weight paths of last passage. This is joint work with Ofer Busani and Timo Seppäläinen