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 transport limited by escape from localized initial states (injection-limited) and transport limited by the extended network (bulk-limited), we identify a 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 distribution with a {strict lower time bound} determined by the properties of the hierarchical network. 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 geometry-encoding function that acts as a diagnostic tool for ascertaining whether or not a given graph is hierarchical.