×

zbMATH — the first resource for mathematics

First passage percolation has sublinear distance variance. (English) Zbl 1087.60070
Summary: Let \(0 < a < b < \infty\), and for each edge e of \(\mathbb Z^d\) let \(\omega_e=a\) or \(\omega_e=b\), each with probability \(1/2\), independently. This induces a random metric \(\text{dist}_\omega\) on the vertices of \(\mathbb Z^d\), called first passage percolation. We prove that for \(d>1\), the distance \(\text{dist}_\omega(0,v)\) 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.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
28A35 Measures and integrals in product spaces
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60E15 Inequalities; stochastic orderings
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] 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. JSTOR: · Zbl 0932.05001
[2] Beckner, W. (1975). Inequalities in Fourier analysis. Ann. of Math. 102 159–182. JSTOR: · Zbl 0338.42017
[3] Bobkov, S. G. and Houdre, C. (1999). A converse Gaussian Poincare-type inequality for convex functions. Statist. Probab. Lett. 44 281–290. · Zbl 0941.60038
[4] Bonami, A. (1970). Etude des coefficients Fourier des fonctiones de \(L^p(G)\). Ann. Inst. Fourier (Grenoble) 20 335–402. · Zbl 0195.42501
[5] 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. · Zbl 0771.60002
[6] Deuschel, J.-D. and Zeitouni, O. (1999). On increasing subsequences of I.I.D. samples. Combin. Probab. Comput. 8 247–263. · Zbl 0949.60019
[7] Durrett, R. (1999). Perplexing problems in probability. Progr. Probab. 44 1–33. · Zbl 0938.01030
[8] Friedgut, E. (200 X ). Influences in product spaces, KKL and BKKKL revisited. · Zbl 1057.60007
[9] Johansson, K. (2000). Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Related Fields 116 445–456. · Zbl 0960.60097
[10] 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.
[11] 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. · Zbl 0602.60098
[12] Kesten, H. (1993). On the speed of convergence in first passage percolation. Ann. Appl. Probab. 3 296–338. JSTOR: · Zbl 0783.60103
[13] Ledoux, M. (2001). The Concentration of Measure Phenomenon . Amer. Math. Soc., Alexandria, VA. · Zbl 0995.60002
[14] Newman, C. M. and Piza, M. (1995). Divergence of shape fluctuations in two dimensions. Ann. Probab. 23 977–1005. JSTOR: · Zbl 0835.60087
[15] 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. · Zbl 0830.60096
[16] Talagrand, M. (1994). On Russo’s approximate zero–one law. Ann. Probab. 22 1576–1587. JSTOR: · Zbl 0819.28002
[17] Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81 73–205. · Zbl 0864.60013
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.