Geodesics and spanning trees for euclidean first-passage percolation

The metric D-alpha(q, q ') on the set Q of particle locations of a homogeneous Poisson process on R-d, defined as the infimum of (Sigmai \q(i) - q(i+1)\ (alpha))(1/alpha) over sequences in Q starting with q and ending with q ' (where \ (.)\ denotes Euclidean distance) has nontrivial geodesics when alpha > 1. The cases 1 < alpha < infinity are the Euclidean first-passage percolation (FPP) models introduced earlier by the authors, while the geodesics in the case alpha = infinity 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 < alpha < infinity (and any d) include inequalities on the fluctuation exponents for the metric (chi less than or equal to 1/2) and for the geodesics (xi less than or equal to 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 less than or equal to alpha < infinity , further results follow concerning spanning trees of semiinfinite geodesics and related random surfaces.