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Explore the infinite bin model (IBM) through this mathematical lecture that examines a family of ranked-biased branching random walks on integers and interacting particle systems. Learn how the IBM is parameterized by probability distributions on positive integers and discover how the speed of the front depends on these distributions. Review special cases including the uniform distribution on finite intervals [1,N] which reduces to branching random walks with selection, and geometric distributions that couple with last passage percolation on complete graphs. Investigate the long memory properties of the IBM, particularly whether sites may reproduce infinitely often. Examine the hydrodynamic limit where explicit computation of front speed becomes possible, revealing wall-crossing phenomena and connections to Dyck paths. The presentation draws from collaborative research with mathematicians from Université Toulouse III Paul Sabatier, CNRS, Université Paris-Saclay, covering both established results and recent developments in this area of statistical mechanics and combinatorics.
