×

zbMATH — the first resource for mathematics

Bigeodesics in first-passage percolation. (English) Zbl 1361.60089
First-passage percolation (FPP) is a model for the flow of a liquid through a random medium due to J. M. Hammersley and D. J. A. Welsh [in: Bernoulli-Bayes-Laplace, Anniversary Vol., Proc. Int. Res. Semin., Berkeley 1963, 61–110 (1965; Zbl 0143.40402)]. On the multi-dimensional integer lattice, every edge \(e \in E^d\) that connects two nearest-neighbor sites of \(\mathbb{Z}^d\) is designated a nonnegative random variable \(t_e\), called the passage time of \(e\). Similarly, for every path \(\gamma\) (from some site to another site), its passage time is defined as \(T (\gamma) := \sum_{e \in \gamma} t_e\). Then the random variable \(T (x, y)\) is defined as the infimum of the passage times of all paths from \(x\) to \(y\). Such a minimizing path is called geodesic and a doubly infinite geodesic is called bigeodesic. Assume that the sequence \((t_e)\) is independent and identically distributed with a continuous distribution. Assume also that \(d=2\). A well-known conjecture of Kensten asserts that there are no bigeodesics in two dimensions almost surely. Let the limit shape be a subset of \(\mathbb{R}^2\) that contains all sites \(x\) such that \(T (0, nx) / n \leq 1\) as \(n \to \infty\) almost surely. Given any deterministic direction \(\phi\) (i.e., for a sequence \((x_n)\) of nearest-neighbor sites in a path, \(\|x_n\|\) tends to infinity and the argument of \(x_n\) tends to \(\phi\) as \(n \to \infty\)) and assuming that the boundary of the limit shape is differentiable, the authors show that there is no bigeodesic with one end directed in \(\phi\) almost surely.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alm, S.E., A note on a problem by welsh in first-passage percolation, Combin. Probab. Comput., 7, 11-15, (1998) · Zbl 0899.60082
[2] Alm, S.E.; Wierman, J., Inequalities for means of restricted first-passage times in percolation theory, Combin. Probab. Comput., 8, 307-315, (1999) · Zbl 0951.60093
[3] Auffinger, A., Damron, M., Hanson, J.: 50 years of first-passage percolation. arXiv:1511.03262 (2015) · Zbl 06828688
[4] Bakhtin, Y.; Cator, E.; Khanin, K., Space-time stationary solutions for the Burgers equation, J. Am. Math. Soc., 27, 193-238, (2014) · Zbl 1296.37051
[5] Benjamini, I.; Kalai, G.; Schramm, O., First passage percolation has sublinear distance variance, Ann. Probab., 31, 1970-1978, (2003) · Zbl 1087.60070
[6] Boivin, D., First passage percolation: the stationary case, Probab. Theory Relat. Fields, 86, 491-499, (1990) · Zbl 0685.60103
[7] Boivin, D.; Derrien, J.-M., Geodesics and recurrence of random walks in disordered systems, Electron. Comm. Probab., 7, 101-115, (2002) · Zbl 1013.60072
[8] Cator, E.; Pimentel, L.P.R., A shape theorem and semi-infinite geodesics for the hammersley model with random weights, ALEA., 8, 163-175, (2011) · Zbl 1276.60109
[9] Coupier, D., Multiple geodesics with the same direction, Electron. Commun. Probab., 16, 517-527, (2011) · Zbl 1244.60093
[10] Coupier, D.; Tran, V., The 2\(D\)-directed spanning forest is almost surely a tree, Random Struct. Algorithms, 42, 59-72, (2012) · Zbl 1257.05159
[11] Damron, M.; Hanson, J., Busemann functions and infinite geodesics in two-dimensional first-passage percolation, Commun. Math. Phys., 325, 917-963, (2014) · Zbl 1293.82014
[12] Ferrari, P.; Pimentel, L.P.R., Competition interfaces and second class particles, Ann. Probab., 33, 1235-1254, (2005) · Zbl 1078.60083
[13] Forgacs, G., Lipowsky, R., Nieuwenhuizen, T.M.: The behaviour of interfaces in ordered and disordered systems. In: C. Domb and J. Lebowitz (eds.) Phase Transitions and Critical Phenomena, Vol. 14, pp. 135-363. Academic Press, London (1991) · Zbl 1087.60070
[14] Georgiou, N., Rassoul-Agha, F., Seppäläinen, T.: Stationary cocycles and Busemann functions for the corner growth model. arXiv:1510.00859 (2015)
[15] Georgiou, N., Rassoul-Agha, F., Seppäläinen, T.: Geodesics and the competition interface for the corner growth model. arXiv:1510.00860 (2015) · Zbl 0951.60093
[16] Häggström, O.; Meester, R., Asymptotic shapes for stationary first passage percolation, Ann. Probab., 23, 1511-1522, (1995) · Zbl 0852.60104
[17] Hoffman, C., Coexistence for Richardson type competing spatial growth models, Ann. Appl. Probab., 15, 739-747, (2005) · Zbl 1067.60098
[18] Hoffman, C., Geodesics in first-passage percolation, Ann. Appl. Probab., 18, 1944-1969, (2008) · Zbl 1153.60055
[19] Howard, C.D., Newman, C.M.: Euclidean models of first-passage percolation. Probab. Theory Relat. Fields 108, 153-170 (1997) · Zbl 0883.60091
[20] Howard, C.D.; Newman, C.M., Geodesics and spanning trees for Euclidean first-passage percolation, Probab. Theory Relat. Fields, 29, 577-623, (2001) · Zbl 1062.60099
[21] Kesten, H.: Aspects of first passage percolation. École d’été de probabilités de Saint-Flour, XIV-1984, Lecture Notes in Math., vol. 1180. Springer, Berlin (1986) · Zbl 0863.60097
[22] Licea, C.; Newman, C.M., Geodesics in two-dimensional first-passage percolation, Ann. Probab., 24, 399-410, (1996) · Zbl 0863.60097
[23] Newman, C.: A surface view of first-passage percolation. In: Proceedings of the International Congress of Mathematicians, Zürich (1994) · Zbl 0848.60089
[24] Newman, C.: Topics in disordered systems. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel. pp. viii+88. ISBN: 3-7643-5777-0 (1997) · Zbl 0897.60093
[25] Pimentel, L.P.R., Multitype shape theorems for FPP models, Adv. Appl. Probab., 39, 53-76, (2007) · Zbl 1112.60083
[26] Tasaki, H., On the upper critical dimensions of random spin systems, J. Stat. Phys., 54, 163-170, (1989)
[27] Wehr, J., On the number of infinite geodesics and ground states in disordered systems, J. Stat. Phys., 87, 439-447, (1997) · Zbl 0937.82020
[28] Wehr, J.; Woo, J., Absence of geodesics in first-passage percolation on a half-plane, Ann. Probab., 26, 358-367, (1998) · Zbl 0937.60092
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.