## Probability in the North East workshop

#### 15 March 2023

Organizers: Thomas Finn (Durham).

**Venue: **MCS0001 in Mathematics and Computer Science Building, Durham University

Attendence is free but registration is **required for organisational purposes**, by emailing Thomas Finn by Wednesday 1st March 2023, also mentioning any special dietary requirements you may have if you want to join for lunch. This event is hosted jointly with RSS Applied Probability section to celebrate Thomas Finn winning the first RSS Applied Probability section biennial PhD Prize.

## Programme

13:00–13:50

Duncan Dauvergne
(University of Toronto)

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.

14:00–14:50

Alexandre Stauffer
(University of Bath)

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).

14:50–15:30

Break

15:30–16:20

Márton Balázs
(University of Bristol)

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