Stochastic drift in discrete waves of non-locally interacting-particles

In this paper, we investigate a generalised model of $N$ particles undergoing second-order non-local interactions on a lattice. Our results have applications across many research areas, including the modelling of migration, information dynamics and Muller's ratchet -- the irreversible accumulation of deleterious mutations in an evolving population. Strikingly, numerical simulations of the model are observed to deviate significantly from its mean-field approximation even for large population sizes. We show that the disagreement between deterministic and stochastic solutions stems from finite-size effects that change propagation speed and cause the position of the wave to fluctuate. These effects are shown to decay anomalously as $(\log N)^2$ and $(\log N)^{3}$, respectively -- much slower than the usual $\sqrt{N}$ factor. As a result, the accumulation of deleterious mutations in a Muller's ratchet and the loss of awareness in a population are processes that occur much faster than predicted by the corresponding deterministic models. The general applicability of our model suggests that this unexpected scaling could be important in a wide range of real-world applications.Comment: 13 pages, 9 figure