When two populations compete for spatially distributed resources, small geometric advantages can produce total dominance. Guha, Ryan, and Karamched model this process by separating resource discovery from resource control, finding that initial symmetry breaking depends on extreme value statistics of first-passage times. A population that is linearly farther from a resource must be exponentially larger to compensate -- the spatial disadvantage translates into a demographic requirement that grows super-linearly.
The mechanism operates through the tail of the first-passage time distribution. The population that discovers a resource first is not the one that is closest on average but the one most likely to produce an extreme early arrival. Because first-passage times have heavy tails in stochastic spatial models, the probability of an early arrival drops sharply with distance, and the number of independent searchers needed to offset this drop scales exponentially. This is a statement about order statistics: the minimum of N random variables from a distribution whose left tail decays exponentially requires N to grow exponentially to shift the minimum by a linear amount.
But discovery alone is insufficient. The study finds that temporary numerical superiority at a resource site cannot sustain dominance without non-reciprocal interaction bias -- an asymmetry in how the two populations affect each other during the control phase. Without this asymmetry, local fluctuations eventually erode early advantages. With it, the system converges to a stable absorbing state where one population controls the entire territory. The interplay between geometric first-passage statistics and interaction asymmetry produces a two-stage mechanism: extreme-value statistics select the initial winner, and non-reciprocity locks in the outcome.
(arXiv:2603.10911)