We introduce a new family of coalescent mean-field interacting particle systems by producing a pinning property that acts over a chosen sequence of multiple time segments. Throughout their evolution, these stochastic particles converge in time (i.e. get pinned) to their random ensemble average at the termination point of any one of the given time segments, only to burst back into life and repeat the underlying principle of convergence in each of the successive time segments, until they are fully exhausted. Although the architecture is represented by a system of piecewise stochastic differential equations, we prove that the conditions generating the pinning property enable every particle to preserve their continuity over their entire lifetime almost surely. As the number of particles in the system increases asymptotically, the system decouples into mutually independent diffusions, which, albeit displaying progressively uncorrelated behaviour, still close in on, and recouple at, a deterministic value at each termination point. Finally, we provide additional analytics including a universality statement for our framework, a study of what we call adjourned coalescent mean-field interacting particles, a set of results on commutativity of double limits, and a proposal of what we call covariance waves.

We study a multivariate system over a finite lifespan represented by a Hermitian-valued random matrix process whose eigenvalues (i) interact in a mean-field way and (ii) converge to their weighted ensemble average at their terminal time. We prove that such a system is guaranteed to converge in time to the identity matrix that is scaled by a Gaussian random variable whose variance is inversely proportional to the dimension of the matrix. As the size of the system grows asymptotically, the eigenvalues tend to mutually independent diffusions that converge to zero at their terminal time, a Brownian bridge being the archetypal example. Unlike commonly studied random matrices that have non-colliding eigenvalues, the proposed eigenvalues of the given system here may collide. We provide the dynamics of the eigenvalue gap matrix, which is a random skew-symmetric matrix that converges in time to the $\textbf{0}$ matrix. Our framework can be applied in producing mean-field interacting counterparts of stochastic quantum reduction models for which the convergence points are determined with respect to the average state of the entire composite system.