Branching random walks and branching Brownian motions are branching systems where each individual, also referred to as particle, is associated with a spatial parameter representing the fitness value of the individual. They are connected to Fisher-Kolmogorov-Petrovskii-Piskunov (F-KPP) partial differential equations, logarithmically correlated Gaussian fields including the two-dimensional Gaussian free field, Liouville Brownian motion, planar quadrangulations, Derrida’s Generalized Random Energy Model (GREM) of spin glasses. I am going to give an elementary and self-contained introduction to the study of the structure of extreme positions in branching random walks and branching Brownian motions.
