Abstract The study of matter fields on an ensemble of random geometries is a difficult problem still in need of new methods and ideas. 
We will follow a point of view inspired by probability theory techniques that relies on an expansion of the two point function as a sum over random walks.
An analogous expansion for Fermions on non-Euclidean geometries is still lacking. Casiday et al. [Laplace and Dirac operators on
graphs , Linear and Multilinear Algebra (2022) 1] proposed a classical “Dirac walk” diffusing on vertices and edges of an oriented graph with a square root of the graph Laplacian. In contrast to the simple random walk, each step of the walk is given a sign depending on the orientation of the edge it goes through. In a toy model, we propose here to study the Green functions, spectrum and the spectral dimension of such “Dirac walks” on the Bethe lattice, a d-regular tree. The recursive structure of the graph makes the problem exactly solvable. Notably, we find that the spectrum develops a gap and that the spectral dimension of the Dirac walk matches that of the simple random walk (ds=1 for d=2 and ds=3 for d ≧3).