High-density hard random geometric graph shortest-path conjecture

Establish whether, in the high-density limit N\to\infty of a hard random geometric graph with N uniformly distributed nodes in a two-dimensional rectangle and link-connectivity distance r_0=O(1/\sqrt{N}), the magnitude of the continuous current density |j(x,y)| induced by current injection at source s and ejection at destination d is proportional, at every point (x,y) of the rectangle, to the probability that (x,y) lies on the shortest path between s and d.

Background

The paper studies the electrical potential and current density generated by opposite point charges, or equivalently current injection and ejection, in a two-dimensional rectangle. It develops exact Green-function representations and efficient q-series formulas for these quantities.

The motivating discrete setting is the hard random geometric graph (HRGG), whose nodes are uniformly distributed in a two-dimensional rectangle and whose edges connect nodes within a prescribed distance r_0. Because the graph is only partially observed through random node positions, the probability that a point belongs to a shortest source-to-destination path is difficult to calculate analytically. The paper conjectures that, when the graph becomes dense under the scaling N\to\infty and r_0=O(1/\sqrt{N}), this discrete shortest-path probability is represented, up to proportionality, by the magnitude of the corresponding continuum current density. The paper does not establish this conjectured correspondence.