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Asymptotics for 2D critical and near-critical first-passage percolation

  • Chang-Long YaoEmail author
Article
  • 23 Downloads

Abstract

We study Bernoulli first-passage percolation (FPP) on the triangular lattice in which sites have 0 and 1 passage times with probability p and \(1-p\), respectively. Denote by \({\mathcal {C}}_{\infty }\) the infinite cluster with 0-time sites when \(p>p_c\), where \(p_c=1/2\) is the critical probability. Denote by \(T(0,{\mathcal {C}}_{\infty })\) the passage time from the origin 0 to \({\mathcal {C}}_{\infty }\). First we obtain explicit limit theorem for \(T(0,{\mathcal {C}}_{\infty })\) as \(p\searrow p_c\). The proof relies on the limit theorem in the critical case, the critical exponent for correlation length and Kesten’s scaling relations. Next, for the usual point-to-point passage time \(a_{0,n}\) in the critical case, we construct subsequences of sites with different growth rate along the axis. The main tool involves the large deviation estimates on the nesting of CLE\(_6\) loops derived by Miller et al. (Ann Probab 44:1013–1052, 2016). Finally, we apply the limit theorem for critical Bernoulli FPP to a random graph called cluster graph, obtaining explicit strong law of large numbers for graph distance.

Keywords

Percolation First passage percolation Correlation length Scaling limit Conformal loop ensemble 

Mathematics Subject Classification

60K35 82B43 

Notes

Acknowledgements

The author thanks Geoffrey Grimmett for an invitation to the Statistical Laboratory in Cambridge University, and thanks the hospitality of the Laboratory, where this work was completed. The author also thanks an anonymous referee for a detailed report that contributed to a better presentation of this manuscript. The author was supported by the National Natural Science Foundation of China (No. 11601505), an NSFC Grant No. 11688101 and the Key Laboratory of Random Complex Structures and Data Science, CAS (No. 2008DP173182).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Academy of Mathematics and Systems ScienceCASBeijingChina

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