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. System state is the empirical distribution of particle locations. The mean-field asymptotic regime, where $n\to\infty$, is considered. We prove that $v_n$, the steady-state speed of the particle system advance, converges, as $n\to\infty$, to a limit $v_{**}$ which can be easily found from a {\em minimum speed selection principle.} Also, as $n\to\infty$, we prove the convergence of the system dynamics 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 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 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 $v$. Using these traveling wave existence results, we provide bounds on an MFL average speed of advance, depending on the right tail exponent of its initial state. We conjecture that these results for exponential jump sizes generalize to general jump sizes.

翻译：我们研究一个由$n$个粒子组成的系统，这些粒子在实线上通过跳跃向前移动。每个粒子既可以进行独立跳跃（跳跃大小服从某种分布），也可以进行“同步”跳跃——即当随机选择的另一个粒子恰好位于其前方时，使其与该粒子会合。系统状态由粒子位置的经验分布描述。我们考虑平均场渐近区域（$n\to\infty$）。证明：粒子系统稳态前进速度$v_n$在$n\to\infty$时收敛于极限$v_{**}$，该极限可通过"最小速度选择原则"轻松求得。同时，当$n\to\infty$时，系统动力学收敛于确定性平均场极限（MFL）。我们证明：任何MFL的平均前进速度均受$v_{**}$下界约束，而由所有粒子初始共位产生的“基准”MFL速度恰好等于$v_{**}$。对于独立跳跃大小服从指数分布的特例，我们证明：存在速度为$v$的行波MFL当且仅当$v\ge v_{**}$，且$v_{**}$具有简洁显式形式；同时证明修正系统（其左边界或右边界以恒定速度$v$移动）中存在行波。利用这些行波存在性结果，我们根据初始状态右尾指数给出MFL平均前进速度的界限。我们推测：这些关于指数跳跃大小的结论可推广至一般跳跃大小分布。