Random walks and ribbon graphs

Based on joint reflections with Andrei Ryabichev. In d-dimensional space we take k steps, starting from point 0. Each step is taken in a random direction by a vector of length 1. It is stated that the average of even powers of the distances by which we move away from the center are integers. I don't know how to explain this yet. But to wander in four-dimensional space, it was necessary to calculate the averages of the products of matrices from the SU2 group. The answer can be written as a sum over ribbon graphs. For d-dimensional walks, you can also use similar pictures.