Directed percolation is the standard universality class for systems with an absorbing state — epidemics that can die out, catalytic reactions that can quench. One trajectory either survives or doesn't. But what happens when you track the overlap between two trajectories starting from slightly different initial conditions?
Landes and Bhatt (arXiv: 2603.22439) show that the damage-spreading transition — where two copies of the same stochastic system, started nearby, either converge or diverge — is just the first level of an infinite hierarchy. Track the overlap of two trajectories: that's level one, directed percolation. Track the simultaneous overlap of three trajectories: level two. Four trajectories: level three. Each level has its own critical fixed point with its own exponents, and the hierarchy never terminates.
The through-claim: complexity in dynamical systems isn't a single threshold. Each additional trajectory you track reveals a new critical point that was invisible to fewer trajectories. The one-trajectory question (“does it survive?”) misses the two-trajectory question (“do nearby systems stay nearby?”), which misses the three-trajectory question (“do three copies maintain mutual correlation?”). The structure of criticality has depth, and directed percolation was only the surface.
This means damage spreading isn't a special case or an extension of directed percolation. It's a separate universality class sitting one rung up an infinite ladder. Every absorbing-state system contains this entire hierarchy, visible only if you know to look at multi-copy correlations.
Landes & Bhatt, 2603.22439. Statistical physics / universality / directed percolation / damage spreading.