Geodesics and Spanning Trees for Euclidean First-Passage Percolation

Simple item page
dc.creatorHoward, C. D.
dc.creatorNewman, C. M.
dc.date2000-10-21
dc.date.accessioned2026-07-07T04:38:09Z
dc.date.available2026-07-07T04:38:09Z
dc.descriptionThe metric $D_α(q,q')$ on the set $Q$ of particle locations of a homogeneous Poisson process on $R^d$, defined as the infimum of $(\sum_i |q_i - q_{i+1}|^α)^{1/α}$ over sequences in $Q$ starting with $q$ and ending with $q'$ (where $| . |$ denotes Euclidean distance) has nontrivial geodesics when $α> 1$. The cases $1 <α< \infty$ are the Euclidean first-passage percolation (FPP) models introduced earlier by the authors while the geodesics in the case $α= \infty$ 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 $1 < α< \infty$ (and any $d$) include inequalities on the fluctuation exponents for the metric ($χ\le 1/2$) and for the geodesics ($ξ\le 3/4$) 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 $2 le α< \infty$, further results follow concerning spanning trees of semi-infinite geodesics and related random surfaces.
dc.description63 pages, one figure; to appear in Ann. Probability
dc.identifierhttps://arxiv.org/abs/math/0010205
dc.identifierhttp://arxiv.org/abs/math/0010205
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/60174
dc.subjectProbability
dc.subjectMathematical Physics
dc.subject60K35, 60G55 (Primary); 82D30, 60F10 (Secondary)
dc.titleGeodesics and Spanning Trees for Euclidean First-Passage Percolation
dc.typetext

Files

Collections

arXiv