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Differentiability and monotonicity of expected passage time in Euclidean first-passage percolation

Part of: Equilibrium statistical mechanics Applications to specific types of physical systems Special processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

C. Douglas Howard
C. Douglas Howard*
Affiliation:
Baruch College, The City University of New York
*
∗ Postal address: Baruch College, The City University of New York, Mathematics Department, 17 Lexington Avenue, New York, NY 10010, USA. Email address: dhoward@baruch.cuny.edu
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Abstract

In first-passage percolation (FPP) models, the passage time Tℓ from the origin to the point ℓeℓ satisfies f(ℓ) := ETℓ = μℓ + o(ℓ½+ε), where μ ∊ (0,∞) denotes the time constant. Yet, for lattice FPP, it is not known rigorously that f(ℓ) is eventually monotonically increasing. Here, for the Poisson-based Euclidean FPP of Howard and Newman (Prob. Theory Relat. Fields108 (1997), 153–170), we prove an explicit formula for f′(ℓ). In all dimensions, for certain values of the model's only parameter we have f′(ℓ) ≥ C > 0 for large ℓ.

Keywords

First-passage percolationpassage timemonotonicity

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B21: Continuum models (systems of particles, etc.) 82B43: Percolation
Secondary: 60F10: Large deviations 82D30: Random media, disordered materials (including liquid crystals and spin glasses)
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 4 , December 2001 , pp. 815 - 827
Copyright
Copyright © by the Applied Probability Trust 2001 

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Footnotes

Research supported in part by NSF Grant DMS-98-15226 and a Eugene Lang Research Fellowship.

References

Alexander, K. S. (1993). A note on some rates of convergence in first-passage percolation. Ann. Appl. Prob. 3, 81–90.Google Scholar
Alm, S. E. (1998). A note on a problem by Welsh in first-passage percolation. Combin. Prob. Comput. 7, 11–15.Google Scholar
Alm, S. E., and Wierman, J. C. (1999). Inequalities for means of restricted first-passage times in percolation theory. Combin. Prob. Comput. 8, 307–315.Google Scholar
Hammersley, J. M., and Welsh, D. J. A. (1965). First-passage percolation, subadditive processes, stochastic networks and generalized renewal theory. In Bernoulli, Bayes, Laplace Anniversary Volume, eds Neyman, J. and LeCam, L., Springer, Berlin, pp. 61–110.Google Scholar
Howard, C. D., and Newman, C. M. (1997). Euclidean models of first-passage percolation. Prob. Theory Relat. Fields 108, 153–170.Google Scholar
Howard, C. D., and Newman, C. M. (1999). From greedy lattice animals to Euclidean first-passage percolation. In Perplexing Problems in Probability, eds Bramson, M. and Durrett, R., Birkhäuser, Boston, pp. 107–119.Google Scholar
Howard, C. D., and Newman, C. M. (2001). Geodesics and spanning trees for Euclidean first-passage percolation. Ann. Prob. 29, 577–623.Google Scholar
Kesten, H. (1993). On the speed of convergence in first-passage percolation. Ann. Appl. Prob. 3, 296–338.Google Scholar
Rudin, W. (1976). Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York.Google Scholar
Serafini, H. C. (1997). First-passage percolation in the Delaunay graph of a d-dimensional Poisson process. Doctoral Thesis, Courant Institute of Mathematical Sciences, New York University.Google Scholar
Vahidi-Asl, M. Q., and Wierman, J. C. (1990). First-passage percolation on the Voronoi tessellation and Delaunay triangulation. In Random Graphs '87, eds Karońske, M., Jaworski, J. and Ruciński, A., John Wiley, New York, pp. 341–359.Google Scholar
Vahidi-Asl, M. Q., and Wierman, J. C. (1992). A shape result for first-passage percolation on the Voronoi tessellation and Delaunay triangulation. In Random Graphs '89, eds Frieze, A. and Luczak, T., John Wiley, New York, pp. 247–262.Google Scholar