Mean-field limits in interacting particle systems with symmetric superlinear rates

Abstract

We derive the exchange-driven growth (EDG) equations as the mean-field limit of interacting particle systems on the complete graph. Extending previous work, we consider symmetric exchange kernels $c(k, l) = c(l, k)$ satisfying super-linear bounds of the form $c(k, l) \leq C(k^{\mu}l^{\nu} + k^{\nu}l^{\mu})$, with $0 \leq \mu, \nu \leq 2$ and $\mu + \nu \leq 3$. Under these conditions the EDG equations are known to have global solutions. We establish a law of large numbers for the empirical measures, showing that the solutions of the EDG equation describe the limiting distribution of cluster sizes in the particle system. Furthermore, we analyse the dynamics of tagged particles and prove convergence to a time-inhomogeneous Markov process governed by a nonlinear master equation, derived via a law of large numbers for size-biased empirical processes.