How dominant is the most efficient pathway through the potential energy landscape of a slowly diffusing disordered system?

It has been suggested that the most-efficient pathway taken by a slowly diffusing many-body system is its geodesic path through the parts of the potential energy landscape lying below a prescribed value of the potential energy. From this perspective, slow diffusion occurs just because these optimal paths become particularly long and convoluted. We test this idea here by applying it to diffusion in two kinds of well-studied low-dimensional percolation problems: the 2d overlapping Lorentz model, and square and simple-cubic bond-dilute lattices. Although the most efficient path should be at its most dominant with the high-dimensional landscapes associated with many-body problems, it is useful to examine simpler, low-dimensional, constant-potential-energy problems such as these ones, both because the simpler models lend themselves to more accurate geodesic-path-finding approaches, and because they offer a significant contrast to many of the models used in the traditional energy-landscape literature. Neither the continuum nor the lattice percolation examples are adequately described by our geodesic-path formalism in the weakly disordered (relatively-fast-diffusion) limit, but in both cases the formalism successfully predicts the existence of the percolation transition and (to a certain extent) the slow diffusion characteristic of near-percolation behavior. The numerical results for these models are not nearly accurate enough near their transitions to describe critical exponents, but the models do showcase the qualitative validity of the geodesic perspective in that they allow us to see explicitly how tortuous and sparse the optimal pathways become as the diffusion constants begin to vanish.