## A Hsu-Robbins-Erdős strong law in first-passage percolation.(English)Zbl 1321.60199

In this paper, a systematic study of the regime for polynomial decay (RPD) of the probability tails is carried out. A precise characterization of the RPD in terms of a moment condition is derived, and, as a consequence, the result improves the statement of the shape theorem without strengthening its hypothesis. In general, the obtained results strengthen earlier strong laws in first-passage percolation, in particular the shape theorem due to D. Richardson [Proc. Camb. Philos. Soc. 74, 515–528 (1973; Zbl 0295.62094)] and J. T. Cox and R. Durrett [Ann. Probab. 9, 583–603 (1981; Zbl 0462.60012)], which states the precise conditions under which the set of points in $$Z^d$$ within distance $$t$$ from the origin, rescaled by $$t$$, converges to a deterministic compact or convex set.

### MSC:

 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60F10 Large deviations 60F15 Strong limit theorems

### Citations:

Zbl 0295.62094; Zbl 0462.60012
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### References:

 [1] Ahlberg, D. (2015). Asymptotics of first-passage percolation on one-dimensional graphs. Adv. in Appl. Probab. 47 182-209. · Zbl 1310.60136 [2] Ahlberg, D. (2015). Convergence towards an asymptotic shape in first-passage percolation on cone-like subgraphs of the integer lattice. J. Theoret. Probab. 28 198-222. · Zbl 1316.60141 [3] Ahlberg, D. (2008). Asymptotics of first-passage percolation on 1-dimensional graphs. Licentiate thesis, Univ. Gothenburg. · Zbl 1310.60136 [4] Antal, P. and Pisztora, A. (1996). On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 1036-1048. · Zbl 0871.60089 [5] Benjamini, I., Kalai, G. and Schramm, O. (2003). First passage percolation has sublinear distance variance. Ann. Probab. 31 1970-1978. · Zbl 1087.60070 [6] Chayes, J. T. and Chayes, L. (1986). Percolation and random media. In Phénomènes Critiques , Systèmes Aléatoires , Théories de Jauge , Part I , II ( Les Houches , 1984) 1001-1142. North-Holland, Amsterdam. · Zbl 0661.60120 [7] Chow, Y. and Zhang, Y. (2003). Large deviations in first-passage percolation. Ann. Appl. Probab. 13 1601-1614. · Zbl 1038.60093 [8] Cox, J. T. and Durrett, R. (1981). Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9 583-603. · Zbl 0462.60012 [9] Cranston, M., Gauthier, D. and Mountford, T. S. (2009). On large deviation regimes for random media models. Ann. Appl. Probab. 19 826-862. · Zbl 1171.60023 [10] Erdös, P. (1949). On a theorem of Hsu and Robbins. Ann. Math. Stat. 20 286-291. · Zbl 0033.29001 [11] Erdös, P. (1950). Remark on my paper “On a theorem of Hsu and Robbins.” Ann. Math. Stat. 21 138. · Zbl 0035.21403 [12] Garet, O. and Marchand, R. (2007). Large deviations for the chemical distance in supercritical Bernoulli percolation. Ann. Probab. 35 833-866. · Zbl 1117.60090 [13] Grimmett, G. and Kesten, H. (1984). First-passage percolation, network flows and electrical resistances. Z. Wahrsch. Verw. Gebiete 66 335-366. · Zbl 0525.60098 [14] Hammersley, J. M. and Welsh, D. J. A. (1965). First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In Proc. Internat. Res. Semin. , Statist. Lab. , Univ. California , Berkeley , Calif. 61-110. Springer, New York. · Zbl 0143.40402 [15] Hsu, P. L. and Robbins, H. (1947). Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33 25-31. · Zbl 0030.20101 [16] Kesten, H. (1986). Aspects of first passage percolation. In École d’Été de Probabilités de Saint-Flour , XIV- 1984. Lecture Notes in Math. 1180 125-264. Springer, Berlin. · Zbl 0602.60098 [17] Kesten, H. (1993). On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3 296-338. · Zbl 0783.60103 [18] Kingman, J. F. C. (1968). The ergodic theory of subadditive stochastic processes. J. Roy. Statist. Soc. Ser. B 30 499-510. · Zbl 0182.22802 [19] Richardson, D. (1973). Random growth in a tessellation. Proc. Cambridge Philos. Soc. 74 515-528. · Zbl 0295.62094 [20] Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81 73-205. · Zbl 0864.60013 [21] Zhang, Y. (2010). On the concentration and the convergence rate with a moment condition in first passage percolation. Stochastic Process. Appl. 120 1317-1341. · Zbl 1195.60128
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