Interacting particle systems are probabilistic models that lie at the “sweet spot” between showing real complexity while still being mathematically tractable. One specifies local random update rules on a lattice or a graph, and out comes global behaviour that is often universal, geometry-dependent, and mathematically deep. In this lecture we focus on particle systems with contact interaction that display surprisingly rich phenomena like nontrivial invariant measures, scaling limits, and genuine phase transitions driven by geometry and dimension. The guiding example is the contact process: a stochastic birth–death dynamics on a graph modeling the spread of an infection (or activity) through nearest-neighbor transmission and spontaneous recovery.
The course focuses on the contact process, emphasizing equivalent representations and the techniques they unlock: the graphical (Harris) construction, duality, couplings and monotonicity, and connections to oriented percolation and renormalization ideas. We develop a rigorous understanding of the subcritical/supercritical regimes, the critical infection rate. Depending on time and the students' preferences, we discuss metastability (e.g. long survival in finite volume and quasi-stationarity) and contrast Euclidean lattices with trees, where non-amenability leads to distinctive phase diagrams (including multiple thresholds).
Branching processes provide a natural model for the growth of populations over time, such as animal populations or the number of infected individuals in a viral epidemic. Simple probabilistic models allow us to understand key features of population dynamics, including exponential growth, extinction probabilities, and critical thresholds. By the end of the course, students will gain a deeper understanding of applied discrete probability and the phase transitions that arise in complex stochastic systems.
