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• W1599386368 abstract "The metric $D_{alpha}(q,q')$ on the set $Q$ of particle locations of a homogeneous Poisson process on $mathbb{R}^d$ , 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 first­passage 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 semiinfinite geodesic has an asymptotic direction and every direction has a semiinfinite geodesic (from every $q$). For $d = 2$ and $2 le alpha < infty$, further results follow concerning spanning trees of semiinfinite geodesics and related random surfaces." @default.
• W1599386368 created "2016-06-24" @default.
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• W1599386368 date "2001-04-01" @default.
• W1599386368 modified "2023-09-25" @default.
• W1599386368 title "Special Invited Paper: Geodesics And Spanning Tees For Euclidean First-Passage Percolaton" @default.
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• W1599386368 doi "https://doi.org/10.1214/aop/1008956686" @default.
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