First passage percolation has sublinear distance variance

Itai Benjamini, Gil Kalai, and Oded Schramm
Source: Ann. Probab. Volume 31, Number 4 (2003), 1970-1978.

Abstract

Let $0 < a < b < \infty$, and for each edge e of $\Z^d$ let $\omega_e=a$ or $\omega_e=b$, each with probability $1/2$, independently. This induces a random metric $\dist_\omega$ on the vertices of $\Z^d$, called first passage percolation. We prove that for $d>1$, the distance $\dist_\omega(0,v)$ from the origin to a vertex $v$, $|v|>2$, has variance bounded by $C|v|/\log|v|$, where $C=C(a,b,d)$ is a constant which may only depend on a, b and d. Some related variants are also discussed.

First Page:
Primary Subjects: 60K35
Secondary Subjects: 60B15, 28A35, 60E15
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.aop/1068646373
Digital Object Identifier: doi:10.1214/aop/1068646373
Mathematical Reviews number (MathSciNet): MR2016607
Zentralblatt MATH identifier: 02077583

References

Baik, J., Deift, P. and Johansson, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119--1178.
Mathematical Reviews (MathSciNet): MR1682248
Digital Object Identifier: doi:10.1090/S0894-0347-99-00307-0
Zentralblatt MATH: 0932.05001
Beckner, W. (1975). Inequalities in Fourier analysis. Ann. of Math. 102 159--182.
Mathematical Reviews (MathSciNet): MR385456
Bobkov, S. G. and Houdre, C. (1999). A converse Gaussian Poincare-type inequality for convex functions. Statist. Probab. Lett. 44 281--290.
Mathematical Reviews (MathSciNet): MR1711613
Bonami, A. (1970). Etude des coefficients Fourier des fonctiones de $L^p(G)$. Ann. Inst. Fourier (Grenoble) 20 335--402.
Mathematical Reviews (MathSciNet): MR283496
Bourgain, J., Kahn, J., Kalai, G., Katznelson, Y. and Linial, N. (1992). The influence of variables in product spaces. Israel J. Math. 77 55--64.
Mathematical Reviews (MathSciNet): MR1194785
Deuschel, J.-D. and Zeitouni, O. (1999). On increasing subsequences of I.I.D. samples. Combin. Probab. Comput. 8 247--263.
Mathematical Reviews (MathSciNet): MR1702546
Digital Object Identifier: doi:10.1017/S0963548399003776
Zentralblatt MATH: 0949.60019
Durrett, R. (1999). Perplexing problems in probability. Progr. Probab. 44 1--33.
Mathematical Reviews (MathSciNet): MR1703122
Zentralblatt MATH: 0938.01030
Friedgut, E. (200X). Influences in product spaces, KKL and BKKKL revisited. Preprint.
Mathematical Reviews (MathSciNet): MR2034300
Digital Object Identifier: doi:10.1017/S0963548303005832
Zentralblatt MATH: 1057.60007
Johansson, K. (2000). Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Related Fields 116 445--456.
Mathematical Reviews (MathSciNet): MR1757595
Digital Object Identifier: doi:10.1007/s004400050258
Zentralblatt MATH: 0960.60097
Kahn, J., Kalai, G. and Linial, N. (1988). The influence of variables on Boolean functions. In Proceedings of the 29th Annual Symposium on Foundations of Computer Science 68--80. IEEE Computer Science Press, Washington, DC.
Kesten, H. (1986). Aspects of first passage percolation. École d'Été de Probabilités de Saint-Flour, XIV. Lecture Notes in Math. 1180 125--264. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR876084
Zentralblatt MATH: 0602.60098
Kesten, H. (1993). On the speed of convergence in first passage percolation. Ann. Appl. Probab. 3 296--338.
Mathematical Reviews (MathSciNet): MR1221154
Ledoux, M. (2001). The Concentration of Measure Phenomenon. Amer. Math. Soc., Alexandria, VA.
Mathematical Reviews (MathSciNet): MR1849347
Zentralblatt MATH: 0995.60002
Newman, C. M. and Piza, M. (1995). Divergence of shape fluctuations in two dimensions. Ann. Probab. 23 977--1005.
Mathematical Reviews (MathSciNet): MR1349159
Pemantle, R. and Peres, Y. (1994). Planar first-passage percolation times are not tight. In Probability and Phase Transition (G. Grimmett, ed.) 261--264. Kluwer, Dordrecht.
Mathematical Reviews (MathSciNet): MR1283187
Zentralblatt MATH: 0830.60096
Talagrand, M. (1994). On Russo's approximate zero--one law. Ann. Probab. 22 1576--1587.
Mathematical Reviews (MathSciNet): MR1303654
Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81 73--205.
Mathematical Reviews (MathSciNet): MR1361756