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Lower bounds for point-to-point wandering exponents in Euclidean first-passage percolation

Published online by Cambridge University Press:  14 July 2016

C. Douglas Howard*
Affiliation:
The City University of New York
*
Postal address: Mathematics Department, Baruch College, The City University of New York, 17 Lexington Ave., New York, NY 10010, USA. Email address: [email protected]

Abstract

In first-passage percolation models, the passage time T(0,L) from the origin to a point L is expected to exhibit deviations of order |L|χ from its mean, while minimizing paths are expected to exhibit fluctuations of order |L|ξ away from the straight line segment . Here, for Euclidean models in dimension d, we establish the lower bounds ξ ≥ 1/(d+1) and χ ≥(1-(d-1)ξ)/2. Combining this latter bound with the known upper bound ξ ≤ 3/4 yields that χ ≥ 1/8 for d=2.

Type
Research Papers
Copyright
Copyright © by the Applied Probability Trust 2000 

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Footnotes

Research supported in part by NSF Grant DMS-98-15226.

References

[1] Baik, J., Deift, P., and Johansson, K. (1999). On the distribution of the longest increasing subsequence in a random permutation. J. Amer. Math. Soc. 12, 11191178.CrossRefGoogle Scholar
[2] 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. 61110.Google Scholar
[3] Howard, C. D., and Newman, C. M. (1997). Euclidean models of first-passage percolation. Prob. Theory Rel. Fields 108, 153170.CrossRefGoogle Scholar
[4] 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, Basel, pp. 107119.Google Scholar
[5] Howard, C. D., and Newman, C. M. (2001). Geodesics and spanning trees for Euclidean first-passage percolation. To appear in Ann. Prob.Google Scholar
[6] Johansson, K. (2000). Transversal fluctuations for increasing subsequences of the plane. Prob. Theory Rel. Fields 116, 445456.CrossRefGoogle Scholar
[7] Licea, C., Newman, C. M., and Piza, M. S. T. (1996). Superdiffusivity in first-passage percolation. Prob. Theory Rel. Fields 106, 559591.Google Scholar
[8] Newman, C. M. (1997). Topics in Disordered Systems. Birkhäuser, Basel.Google Scholar
[9] Newman, C. M., and Piza, M. S. T. (1995). Divergence of shape fluctuations in two dimensions Ann. Prob. 23, 9771005.Google Scholar
[10] Serafini, H. C. (1997). First-passage percolation in the Delaunay graph of a d-dimensional Poisson process. PhD Thesis, Courant Institute of Mathematical Sciences, New York University.Google Scholar
[11] 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. 341359.Google Scholar
[12] 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. 247262.Google Scholar
[13] Wüthrich, M. V. (1998). Fluctuation results for Brownian motion in a Poissonian potential. Ann. Inst. H. Poincaré Prob. Statist. 34, 279308.CrossRefGoogle Scholar
[14] Wüthrich, M. V. (1998). Superdiffusive behavior of two-dimensional Brownian motion in a Poissonian potential. Ann. Prob. 26, 10001015.Google Scholar
[15] Wüthrich, M. V. (1998). Scaling identity for crossing Brownian motion in a Poissonian potential. Prob. Theory Rel. Fields 112, 299319.CrossRefGoogle Scholar