Quantum Networks and Classical Self-Avoiding Random Walks

In a quantum network model, unitary matrices are assigned to each edge and node of a graph.  The quantum amplitude for a particle to propagate from node A to node B is the sum over all random walks (Feynman paths) from A to B, each walk being weighted by the ordered product of matrices along the path.  In most cases these models are too difficult to solve analytically, but I shall argue that when the matrices are random elements of SU("), independently drawn from the invariant measure on that group, then averages of these quantum amplitudes are equal to the probability that a certain kind of self-avoiding *classical* random walk reaches B when started at A.  This leads to various conjectures about the generic behaviour of such network models on regular lattices in two and three dimensions.