zbMATH — the first resource for mathematics

Lower bounds for fluctuations in first-passage percolation for general distributions. (English. French summary) Zbl 1434.60280
Summary: In first-passage percolation (FPP), one assigns i.i.d. weights to the edges of the cubic lattice \(\mathbb{Z}^d\) and analyzes the induced weighted graph metric. If \(T(x,y)\) is the distance between vertices \(x\) and \(y\), then a primary question in the model is: what is the order of the fluctuations of \(T(0,x)\)? It is expected that the variance of \(T(0,x)\) grows like the norm of \(x\) to a power strictly less than 1, but the best lower bounds available are (only in two dimensions) of order \(\log \|x\|\). This result was found in the ’90s and there has not been any improvement since. In this paper, we address the problem of getting stronger fluctuation bounds: to show that \(T(0,x)\) is with high probability not contained in an interval of size \(o(\log \|x\|)^{1/2} \), and similar statements for FPP in thin cylinders. Such statements have been proved for special edge-weight distributions, and here we obtain such bounds for general edge-weight distributions. The methods involve inducing a fluctuation in the number of edges in a box whose weights are of “hi-mode” (large).

60K35 Interacting random processes; statistical mechanics type models; percolation theory
Full Text: DOI Euclid
[1] M. Aizenman, F. Germinet, A. Klein and S. Warzel. On Bernoulli decompositions for random variables, concentration bounds, and spectral localization. Probab. Theory Related Fields 143 (2009) 219-238. · Zbl 1152.60020
[2] K. Alexander. Approximation of subadditive functions and convergence rates in limiting-shape results. Ann. Probab. 25 (1997) 30-55. · Zbl 0882.60090
[3] A. Auffinger and M. Damron. Differentiability at the edge of the percolation cone and related results in first-passage percolation. Probab. Theory Related Fields 156 (2013) 193-227. · Zbl 1275.60093
[4] A. Auffinger, M. Damron and J. Hanson. Rate of convergence of the mean for sub-additive ergodic sequences. Adv. Math. 285 (2014) 138-181. · Zbl 1334.60047
[5] A. Auffinger, M. Damron and J. Hanson. 50 Years of First-Passage Percolation. University Lecture Series 68. American Mathematical Society, Providence, RI, 2017. v+161 pp. · Zbl 1452.60002
[6] E. Bates and S. Chatterjee. Fluctuation lower bounds in planar random growth models, 2018. Available at arXiv:1810.03656.
[7] M. Benaïm and R. Rossignol. Exponential concentration for first passage percolation through modified Poincaré inequalities. Ann. Inst. Henri Poincaré B, Probab. Stat. 44 (2008) 544-573. · Zbl 1186.60102
[8] I. Benjamini, G. Kalai and O. Schramm. First passage percolation has sublinear distance variance. Ann. Probab. 31 (2003) 1970-1978. · Zbl 1087.60070
[9] R. Cerf and M. Théret. Weak shape theorem in first passage percolation with infinite passage times. Ann. Inst. Henri Poincaré B, Probab. Stat. 52 (2016) 1351-1381. · Zbl 1350.60100
[10] S. Chatterjee. The universal relation between scaling exponents in first-passage percolation. Ann. of Math. (2) 177 (2013) 1-35. · Zbl 1271.60101
[11] S. Chatterjee. A general method for lower bounds on fluctuations of random variables. Ann. Probab. (2017). To appear.
[12] S. Chatterjee and P. Dey. Central limit theorem for first-passage percolation time across thin cylinders. Probab. Theory Related Fields 156 (2013) 613-663. · Zbl 1274.60287
[13] M. Damron, J. Hanson and P. Sosoe. Sublinear variance in first-passage percolation for general distributions. Probab. Theory Related Fields 163 (2015) 223-258. · Zbl 1331.60182
[14] M. Damron, W.-K. Lam and X. Wang. Asymptotics for \(2D\) critical first-passage percolation. Ann. Probab. 45 (2017) 2941-2970. · Zbl 1378.60115
[15] R. Durrett. Oriented percolation in two dimensions. Ann. Probab. 12 (1984) 999-1040. · Zbl 0567.60095
[16] R. Gong, C. Houdré and J. Lember. Lower bounds on the generalized central moments of the optimal alignments score of random sequences. J. Theoret. Probab. 31 (2018) 643-683. · Zbl 1409.60026
[17] G. Grimmett Percolation, 2nd edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer-Verlag, Berlin, 1999. xiv+444 pp.
[18] C. Houdré and J. Ma. On the order of the central moments of the length of the longest common subsequence in random words. In High Dimensional Probability VII 105-136. Progr. Probab. 71. Springer, Cham, 2016. · Zbl 1382.60020
[19] C. Houdré and C. Xu. Concentration of geodesics in directed Bernoulli percolation, 2018. Available at arXiv:1607.02219.
[20] C. Houdré and C. Xu. Power low bounds for the central moments of the last passage time for directed percolation in a thin rectangle, 2018. Available at arXiv:1905.10034.
[21] K. Johansson. Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000) 437-476. · Zbl 0969.15008
[22] H. Kesten. Aspects of first passage percoclation. In École d’Été de Probabilités de Saint Flour XIV 125-264. Lecture Notes in Mathematics 1180, 1986.
[23] H. Kesten. On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3 (1993) 296-338. · Zbl 0783.60103
[24] J. Lember and H. Matzinger. Standard deviation of the longest common subsequence. Ann. Probab. 37 (2009) 1192-1235. · Zbl 1182.60004
[25] C. Licea, C. M. Newman and M. Piza. Superdiffusivity in first-passage percolation. Probab. Theory Related Fields 106 (1996) 559-591. · Zbl 0870.60096
[26] R. Marchand. Strict inequalities for the time constant in first passage percolation. Ann. Appl. Probab. 12 (2002) 1001-1038. · Zbl 1062.60100
[27] C. M. Newman and M. Piza. Divergence of shape fluctuations in two dimensions. Ann. Probab. 23 (1995) 977-1005. · Zbl 0835.60087
[28] R. Pemantle and Y. Peres. Planar first-passage percolation times are not tight. In Probability and Phase Transition (Cambridge, 1993) 261-264. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 420. Kluwer Acad. Publ., Dordrecht, 1994. · Zbl 0830.60096
[29] E. Sperner. Ein satz über Untermengen einer endlichen Menge. Math. Z. 27 (1928) 544-548. · JFM 54.0090.06
[30] R. Tessera. Speed of convergence in first passage percolation and geodesicity of the average distance. Ann. Inst. Henri Poincaré B, Probab. Stat. 54 (2014) 569-586. · Zbl 1390.60357
[31] C. Xu. Topics in percolation and sequence analysis. Ph. D. Thesis, Georgia Institute of Technology, 2018.
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.