Geodesics in two-dimensional first-passage percolation
Cristina Licea and Charles M. Newman
Source: Ann. Probab. Volume 24, Number 1
(1996), 399-410.
Abstract
We consider standard first-passage percolation on $\mathbb{Z}^2$. Geodesics are nearest-neighbor paths in $\mathbb{Z}^2$, each of whose segments is time-minimizing. We prove part of the conjecture that doubly infinite geodesics do not exist. Our main tool is a result of independent interest about the coalescing of semi-infinite geodesics.
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Permanent link to this document: http://projecteuclid.org/euclid.aop/1042644722
Mathematical Reviews number (MathSciNet): MR1387641
Digital Object Identifier: doi:10.1214/aop/1042644722
Zentralblatt MATH identifier: 0863.60097
References
1 BURTON, R. and KEANE, M. 1989. Density and uniqueness in percolation. Comm. Math. Phy s. 121 501 505.
Mathematical Reviews (MathSciNet): MR90g:60090
Zentralblatt MATH: 0662.60113
Digital Object Identifier: doi:10.1007/BF01217735
Project Euclid: euclid.cmp/1104178143
2 FORGACS, G., LIPOWSKY, R. and NIEUWENHUIZEN, T. M. 1991. The behaviour of interfaces in Z ordered and disordered sy stems. In Phase Transitions and Critical Phenomena C.. Domb and J. Lebowitz, eds. 14 135 363. Academic, London.
3 HAMMERSLEY, J. M. and WELSH, D. J. A. 1965. First-passage percolation, subadditive processes, stochastic networks and generalized renewal theory. In Bernoulli, Bay es, Z. Laplace Anniversary Volume J. Ney man and L. Le Cam, eds. 61 110. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR33:6731
Zentralblatt MATH: 0143.40402
4 KESTEN, H. 1986. Aspects of first-passage percolation. Lecture Notes in Math. 1180 125 264. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR876084
Zentralblatt MATH: 0602.60098
5 KESTEN, H. 1987. Percolation theory and first-passage percolation. Ann. Probab. 15 1231 1271.
Mathematical Reviews (MathSciNet): MR88g:60246
Zentralblatt MATH: 0629.60103
Digital Object Identifier: doi:10.1214/aop/1176991975
Project Euclid: euclid.aop/1176991975
6 NEWMAN, C. M. 1995. A surface view of first-passage percolation. Proc. 1995 Internat. Cong. Z. Math. S. D. Chatterji, ed. 1017 1023. Birkhauser, Boston. ¨
7 NEWMAN, C. M. and SCHULMAN, L. S. 1981. Infinite clusters in percolation models. J. Statist. Phy s. 26 613 628.
Zentralblatt MATH: 0509.60095
Mathematical Reviews (MathSciNet): MR648202
Digital Object Identifier: doi:10.1007/BF01011437
8 WEHR, J. 1995. Private communication.
NEW YORK, NEW YORK 10004-1064 NEW YORK UNIVERSITY 251 MERCER STREET
NEW YORK, NEW YORK 10012 E-mail: newman@cims.ny u.edu
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