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 $(\sum_i |q_i - q_{i+1}|^\alpha)^{1/\alpha}$ over sequences in $Q$ starting with $q$ and ending with $q'$ (where $| . |$ denotes Euclidean distance) has nontrivial geodesics when $\alpha > 1$. The cases $1 <\alpha < \infty$ are the Euclidean first-passage percolation (FPP) models introduced earlier by the authors while the geodesics in the case $\alpha = \infty$ 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 $1 < \alpha < \infty$ (and any $d$) include inequalities on the fluctuation exponents for the metric ($\chi \le 1/2$) and for the geodesics ($\xi \le 3/4$) in strong enough versions to yield conclusions not yet obtained for lattice FPP: almost surely, every semi-infinite geodesic has an asymptotic direction and every direction has a semi-infinite geodesic (from every $q$). For $d=2$ and $2 le \alpha < \infty$, further results follow concerning spanning trees of semi-infinite geodesics and related random surfaces.