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, or ``synchronization'' jumps, which allow it to join a randomly chosen other particle if the latter happens to be ahead of it. The mean-field asymptotic regime, where $n\to\infty$, is considered. As $n\to\infty$, we prove the convergence of the system dynamics to that of a deterministic mean-field limit (MFL). We obtain results on the average speed of advance of a ``benchmark'' MFL (BMFL) and the liminf of the steady-state speed of advance, in terms of MFLs that are traveling waves. For 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 simple explicit form; this allows us to show that the average speed of the BMFL is equal to $v_*$ and the liminf of the steady-state speeds is lower bounded by $v_*$. Finally, we put forward a conjecture that both the average speed of the BMFL and the exact limit of the steady-state speeds, under general distribution of an independent jump size, are equal to number $v_{**}$, which is easily found from a ``minimum speed principle.'' This general conjecture is consistent with our results for the exponentially distributed jumps and is confirmed by simulations.

翻译：我们研究一个由 $n$ 个粒子组成的系统，这些粒子在实线上以跳跃方式向前运动。每个粒子既可以进行独立跳跃（跳跃大小遵循某种分布），也可以进行“同步”跳跃——当随机选取的另一粒子位于其前方时，该跳跃允许其追上该粒子。考虑平均场渐近区域，即 $n\to\infty$ 的情况。当 $n\to\infty$ 时，我们证明系统动力学收敛到确定的平均场极限（MFL）。我们通过行波MFL，获得了“基准”平均场极限（BMFL）的平均前进速度以及稳态前进速度的下极限结果。对于独立跳跃大小服从指数分布的特例，我们证明当且仅当 $v\ge v_*$ 时存在速度为 $v$ 的行波MFL，其中 $v_*$ 具有简单的显式形式；由此可证BMFL的平均速度等于 $v_*$，且稳态速度的下极限以 $v_*$ 为下界。最后，我们提出猜想：在独立跳跃大小的一般分布下，BMFL的平均速度与稳态速度的精确极限均等于可通过“最小速度原理”简单求得的数值 $v_{**}$。该一般性猜想与指数分布跳跃下的结论一致，并得到了模拟结果的验证。