Special Invited Paper: Geodesics And Spanning Tees For Euclidean First-Passage Percolaton

The metric Dα(q,q') on the set Q of particle locations of a homogeneous Poisson process on Rᵈ , defined as the infimum of (∑ᵢ |qᵢ - qᵢ₊₁|α)1/α over sequences in Q starting with q and ending with $q'$ (where $|·|$ denotes Euclidean distance) has nontrivial geodesics when α>1. The cases 1< α<∞ are the Euclidean first­passage percolation (FPP) models introduced earlier by the authors, while the geodesics in the case α = ∞ are exactly the paths from the Euclidean minimal spanning trees/forests of Aldous and Steele. We compare and contrast results and conjectures for these two situations. New results for 1 < α < ∞ (and any d) include inequalities on the fluctuation exponents for the metric (χ ≤ 1/2) and for the geodesics (ξ ≤ 3/4) in strong enough versions to yield conclusions not yet obtained for lattice FPP: almost surely, every semiinfinite geodesic has an asymptotic direction and every direction has a semiinfinite geodesic (from every q). For $d = 2$ and 2 ≤ α < ∞, further results follow concerning spanning trees of semiinfinite geodesics and related random surfaces.