×

zbMATH — the first resource for mathematics

A simplified proof of the relation between scaling exponents in first-passage percolation. (English) Zbl 1296.60257
The Chatterjee result provides a relation between the transversal exponent and the fluctuation exponent in first-passage multidimensional percolation. The basic idea of Chatterjee is to use a nearly Gamma assumption on the passage time, and here one shows how the same result can be obtained without this assumption.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid
References:
[1] Alexander, K. S. (1997). Approximation of subadditive functions and convergence rates in limiting-shape results. Ann. Probab. 25 30-55. · Zbl 0882.60090
[2] Auffinger, A. and Damron, M. (2013). The scaling relation \(\chi=2\xi-1\) for directed polymers in a random environment. · Zbl 1277.82029
[3] Benaïm, M. and Rossignol, R. (2008). Exponential concentration for first passage percolation through modified Poincaré inequalities. Ann. Inst. Henri Poincaré Probab. Stat. 44 544-573. · Zbl 1186.60102
[4] Blair-Stahn, N. D. (2010). First passage percolation and competition models. Available at . 1005.0649
[5] Chatterjee, S. (2013). The universal relation between scaling exponents in first-passage percolation. Ann. of Math. (2) 177 663-697. · Zbl 1271.60101
[6] Chatterjee, S. and Dey, P. S. (2013). Central limit theorem for first-passage percolation time across thin cylinders. Probab. Theory Related Fields 156 613-663. · Zbl 1274.60287
[7] Durrett, R. and Liggett, T. M. (1981). The shape of the limit set in Richardson’s growth model. Ann. Probab. 9 186-193. · Zbl 0457.60083
[8] Howard, C. D. (2004). Models of first-passage percolation. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 125-173. Springer, Berlin. · Zbl 1206.82048
[9] Johansson, K. (2000). Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Related Fields 116 445-456. · Zbl 0960.60097
[10] Kardar, M. and Zhang, Y. C. (1987). Scaling of directed polymers in random media. Phys. Rev. Lett. 56 2087-2090.
[11] Kesten, H. (1993). On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3 296-338. · Zbl 0783.60103
[12] Kesten, H. (2003). First-passage percolation. In From Classical to Modern Probability. Progress in Probability 54 93-143. Birkhäuser, Basel. · Zbl 1041.60077
[13] Krug, J. (1987). Scaling relation for a growing surface. Phys. Rev. A (3) 36 5465-5466.
[14] Licea, C., Newman, C. M. and Piza, M. S. T. (1996). Superdiffusivity in first-passage percolation. Probab. Theory Related Fields 106 559-591. · Zbl 0870.60096
[15] Newman, C. M. and Piza, M. S. T. (1995). Divergence of shape fluctuations in two dimensions. Ann. Probab. 23 977-1005. · Zbl 0835.60087
[16] Wehr, J. and Aizenman, M. (1990). Fluctuations of extensive functions of quenched random couplings. J. Stat. Phys. 60 287-306. · Zbl 0718.60129
[17] Wüthrich, M. V. (1998). Scaling identity for crossing Brownian motion in a Poissonian potential. Probab. Theory Related Fields 112 299-319. · Zbl 0938.60099
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.