Coalescence on the discrete torus revisited

Here, we provide an alternative and shorter proof of a theorem by Beltrán et al. (J Stat Phys 177:1172–1206) regarding the convergence of the number of particles in the coalescent process on the discrete torus to the number of partitions in Kingman’s coalescent. The proof relies heavily on the work of Cox (Ann Probab 17:1333–1366, 1989) on the coalescing process in the discrete torus. Using some of Cox’s estimates, we directly prove the convergence of the finite-dimensional distributions of the processes involved. This, together with the tightness of the process, leads us to our result.