A down-up chain with persistent labels on multifurcating trees

In this article, we propose to study a general notion of a down-up Markov chain for multifurcating trees with $n$ labeled leaves. We study in detail down-up chains associated with the $(\alpha ,\gamma )$-model of Chen et al. (Electron. J. Probab. 14 (2009), 400–430.), generalizing and further developing previous work by Forman et al. (arXiv:1802.00862, 2018; arXiv:1804.01205, 2018; arXiv:1809.07756, 2018; Random Struct. Algoritm. 54 (2020), 745–769; Electron. J. Probab. 25 (2020), 1–46.) in the binary special cases. The technique we deploy utilizes the construction of a growth process and a down-up Markov chain on trees with planar structure. Our construction ensures that natural projections of the down-up chain are Markov chains in their own right. We establish label dynamics that at the same time preserve the labeled alpha-gamma distribution and keep the branch points between the $k$ smallest labels for order $n^2$ time steps for all $k\ge 2$. We conjecture the existence of diffusive scaling limits generalizing the “Aldous diffusion” by Forman et al. (arXiv:1802.00862, 2018; arXiv:1804.01205, 2018; arXiv:1809.07756, 2018.) as a continuum-tree-valued process and the “algebraic $\alpha$-Ford tree evolution” by Löhr et al. (Ann. Probab. 48 (2020), 2565–2590.) and by Nussbaumer and Winter (arXiv:2006.09316, 2020.) as a process in a space of algebraic trees.