A large-scale particle system with independent jumps and distributed synchronization

Abstract We study a system consisting of n particles, moving forward in jumps on the real line. Each particle can make both independent jumps, whose sizes have some distribution, and ‘synchronization’ jumps, which allow it to join a randomly chosen other particle if the latter happens to be ahead of it. The system state is the empirical distribution of particle locations. We consider the mean-field asymptotic regime where $n\to\infty$ . We prove that $v_n$ , the steady-state speed of advance of the particle system, converges, as $n\to\infty$ , to a limit $v_{**}$ which can easily be found from a minimum speed selection principle . Also we prove that as $n\to\infty$ , the system dynamics converges to that of a deterministic mean-field limit (MFL). We show that the average speed of advance of any MFL is lower-bounded by $v_{**}$ , and the speed of a ‘benchmark’ MFL, resulting from all particles initially being co-located, is equal to $v_{**}$ . In the special case of exponentially distributed independent jump sizes, we prove that a traveling-wave MFL with speed v exists if and only if $v\ge v_{**}$ , with $v_{**}$ having a simple explicit form; we also show the existence of traveling waves for the modified systems with a left or right boundary moving at a constant speed v . We provide bounds on an MFL’s average speed of advance, depending on the right tail exponent of its initial state. We conjecture that these results for exponential jump sizes extend to general jump sizes.