Laplacian-based random walks on simplicial complexes with analyzable limiting behavior

Develop Laplacian-based random walk frameworks on general simplicial complexes that are topologically informative and admit rigorously analyzable limiting behavior, including clear conditions for convergence and long-term dynamics.

Background

Random walks generated by graph Laplacians are irreducible on connected graphs and have well-understood limit behavior, but analogous constructions on higher-dimensional simplicial complexes typically lack irreducibility and exhibit more complex dynamics.

The authors call for Laplacian-based random walks on simplicial complexes that both reflect topological structure and have limiting behavior that can be rigorously analyzed, addressing current gaps in understanding the stochastic dynamics induced by higher-order Laplacians.

References

Other open problems include defining topologically informative Laplacian-based random walks on simplicial complexes that their limiting behavior could be easily analyzed, or extending Morse–Floer-type constructions to broader classes of dynamical systems.

— From the discrete to the continuous, from simplicial complexes to Riemannian manifolds. Approximating flows and cuts on manifolds by discrete versions (2512.05319 - Eidi et al., 4 Dec 2025) in Section 6 (Conclusion and Outlook)