Simplicial branching random walks

We study a model of branching random walks on simplicial complexes, which can be seen as a natural generalization of random walks on graphs. Exploiting the fact that each of its particles is distributed like the (d-1)-walk introduced in Parzanchevski and Rosenthal (Random Struct Algorithms 50(2):225–261, 2017), we show the model is connected to the spectral and topological properties of the complex. The branching model is then used in order to calculate the spectral measure of the upper Laplacian associated with high-dimensional analogues of regular trees, thus obtaining a Kesten–McKay type distribution in arbitrary dimensions. Finally, we use the branching model in order to construct solutions to the Dirichlet problem on simplicial complexes.