Diffusive scaling limits for multifurcating-tree Markov chains

Establish diffusive scaling limits for the Markov chains on unranked, labeled, planar multifurcating trees introduced by Sørensen, thereby generalizing the Aldous diffusion to a continuum-tree-valued process.

Background

The paper notes that studies of Markov chains on multifurcating trees are scarce. It cites Sørensen (2024), who defines down-up chains with persistent labels on unranked, labeled, planar multifurcating trees and conjectures that these chains admit diffusive scaling limits.

The unresolved issue is to prove the existence and characterize the limiting continuum-tree-valued process, extending the Aldous diffusion known from related binary-tree settings. This problem is mentioned as part of the broader mathematical context rather than addressed by the lattice construction developed in the paper.

References

define Markov chains on unranked, labeled, planar multifurcating trees based on the $(\alpha, \gamma)$ splitting tree \citep{haas2008, chen2009} and conjecture the existence of diffusive scaling limits generalizing the ``Aldous diffusion'' as a continuum-tree-values process.

— The space of multifurcating ranked tree shapes: enumeration, lattice structure, and Markov chains  (2506.10856 - Zhang et al., 12 Jun 2025) in Section 1, Introduction