Exact collective first-passage statistics of N trail-interacting walkers

What the math gives to ML

The paper provides an exact mechanism for collective dynamics with shared environmental memory: when the trail response saturates, the distribution of order statistics of boundary-arrival times is invariant to whether walkers evolve simultaneously or sequentially, whereas nonsaturating responses retain scheduling dependence. It also derives a scale-invariant form for the kth absorption-time distribution, F_k,N(t) = t^(-1) f_k,N(x_1,0^(d_w)/t, ..., x_N,0^(d_w)/t), implying measurable persistence exponents. A transferable neural-network construction is a bounded shared-memory field for sets of recurrent agents, with asynchronous execution as a computational approximation whose outputs should match parallel execution once the memory response enters its saturation regime.