The Annals of Probability

Special Invited Paper: Geodesics And Spanning Tees For Euclidean First­Passage Percolaton

C. Douglas Howard and Charles M. Newman

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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.

Article information

Source
Ann. Probab., Volume 29, Number 2 (2001), 577-623.

Dates
First available in Project Euclid: 21 December 2001

Permanent link to this document
https://projecteuclid.org/euclid.aop/1008956686

Digital Object Identifier
doi:10.1214/aop/1008956686

Mathematical Reviews number (MathSciNet)
MR1849171

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G55: Point processes
Secondary: 82D30: Random media, disordered materials (including liquid crystals and spin glasses) 60F10: Large deviations

Keywords
first­passage percolation random metric minimal spanning tree geodesic combinatorial optimization shape theorem random surface Poisson process

Citation

Howard, C. Douglas; Newman, Charles M. Special Invited Paper: Geodesics And Spanning Tees For Euclidean First­Passage Percolaton. Ann. Probab. 29 (2001), no. 2, 577--623. doi:10.1214/aop/1008956686. https://projecteuclid.org/euclid.aop/1008956686


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