Entropic Collapse and Extreme First-Passage Times in Discrete Ballistic Transport

Abstract: We investigate the extreme first-passage statistics of N non-interacting random walkers on discrete, hierarchical networks. By distinguishing between "injection-limited" and "bulk-limited" transport, we identify a novel universality class of extreme value statistics that arises in geometries dominated by source traps (e.g., the Comet graph). In this regime, the distribution of the minimum arrival time does not converge to any of the classical generalized extreme value distributions. Instead, it follows a discrete, hard-edge distribution determined by the properties of the source. We analytically derive the asymptotic behavior of this class and validate our predictions against Monte Carlo simulations. Crucially, we identify the mechanism of "entropic collapse" that destroys this scaling in bulk-dominated geometries like the Bethe lattice, where the phase space of delayed paths diverges with distance. This work establishes a rigorous topological condition -- the invariance of the entropic penalty for detours -- that separates quantum-like, discrete extreme value statistics from classical bulk transport limits.