Planar graph navigation is an important problem with significant implications to both point location in geometric data structures and routing in networks. However, whilst a number of algorithms and existence proofs have been proposed, very little analysis is available for the properties of the paths generated and the computational resources required to generate them under a random distribution hypothesis for the input. In this paper we analyse a new deterministic planar navigation algorithm with constant competitiveness which follows vertex adjacencies in the Delaunay triangulation. We call this strategy \emph{cone walk}. We prove that given n uniform points in a smooth convex domain of unit area, and for any start point z and query point q; cone walk applied to z and q will access at most O(|zq| √n + log^7 n) sites with complexity O(|zq| √n log log n + log^7 n) with probability tending to 1 as n goes to infinity. We additionally show that in this model, cone walk is (log^(3+x) n)-memoryless with high probability for any pair of start and query point in the domain, for any positive x. We take special care throughout to ensure our bounds are valid even when the query points are arbitrarily close to the border.