The Singular Catastrophe

In a branching random walk, particles reproduce and move. Each generation, individuals spawn offspring who take random steps. The population spreads through space like a growing cloud. When the step distribution has light tails — Gaussian or exponential — extreme deviations result from the accumulation of many moderate steps along a lineage. The trajectory to the extreme is gradual.

Stonner (arXiv:2603.16316) proves that when the step distribution has heavy tails, the mechanism inverts entirely. An extreme deviation — finding a particle far from where the population should be — is caused almost surely by a single enormous jump somewhere in the genealogy. One ancestor, one step. The probability of the extreme event equals the tail probability of one individual step, multiplied by the expected population size. Catastrophe is singular, not cumulative.

This is the “principle of one big jump” extended to branching processes. In the classical (non-branching) setting, it was known that heavy-tailed random walks reach extremes through single jumps rather than accumulated drift. The branching adds a complication: more individuals means more chances for the big jump, and the population structure determines where in the genealogy the jump is most likely to occur. But the fundamental structure persists — one event, one cause.

The principle divides the world into two regimes. In the light-tailed regime, extremes are democratic: many small contributions from many ancestors. In the heavy-tailed regime, extremes are autocratic: one contribution from one ancestor. The genealogy of the extreme event, traced backward, finds a single point of origin rather than a distributed cause.

When tails are heavy, narrative causation — this led to that led to the other — is a Gaussian illusion. The right question is not “what chain of events produced this?” but “which single event was it?”