Torus Walk

This tool allows users to see the higher-dimensional analogues of random walks on graphs. Instead of

• a graph, we use a CW complex,
• a random walk on the vertices, we use a random walk on the edge (1-cells)
• a particle or point walking, we use a loop (1-cycle)

The physical process which governs the random walk has two key components: a deterministic part, which is the gradient flow of a function defined on the edges, and a stochastic part, which is a random perturbation. The deterministic part causes the loop to rotate, while the stochastic part causes it to expand randomly at each time step. The balance between these two forces determines the behavior of the walk.

Note that at all times, the evolved loop is homologous to the initial condition (loop).