Abstract
We study first-passage percolation in two dimensions, using measures μ on passage times with b: = inf supp(μ) > 0 and \({\mu(\{b\})=p\geq \vec p_c}\) , the threshold for oriented percolation. We first show that for each such μ, the boundary of the limit shape for μ is differentiable at the endpoints of flat edges in the so-called percolation cone. We then conclude that the limit shape must be non-polygonal for all of these measures. Furthermore, the associated Richardson-type growth model admits infinite coexistence and if μ is not purely atomic the graph of infection has infinitely many ends. We go on to show that lower bounds for fluctuations of the passage time given by Newman–Piza extend to these measures. We establish a lower bound for the variance of the passage time to distance n of order log n in any direction outside the percolation cone under a condition of finite exponential moments for μ. This result confirms a prediction of Newman and Piza (Ann Probab 23:977–1005, 1995) and Zhang (Ann Probab 36:331–362, 2008). Under the assumption of finite radius of curvature for the limit shape in these directions, we obtain a power-law lower bound for the variance and an inequality between the exponents χ and ξ.
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Alexander K.: Approximation of subadditive functions and convergence rates in limiting-shape results. Ann. Probab. 1, 30–55 (1997)
Blair-Stahn, N.D.: First passage percolation and competition models (2010). arXiv:1005.0649
Chatterjee, S., Dey, P.: Central limit theorem for first-passage percolation time across thin cylinders (2009). arXiv:0911.5702
Cox J.T., Durrett R.: Some limit theorems for percolation with necessary and sufficient conditions. Ann. Probab. 9, 583–603 (1981)
Damron, M., Hochman, M.: Examples of non-polygonal limit shapes in i.i.d. first-passage percolation and infinite coexistence in spatial growth models (2010). arXiv:1009.2523
Durrett R.: Oriented percolation in two dimensions. Ann. Probab. 12, 999–1040 (1984)
Durrett R., Liggett T.: The shape of the limit set in Richardson’s growth model. Ann. Probab. 9, 186–193 (1981)
Eden, M.: A two-dimensional growth process. In: Proc. 4th Berkeley Sympos. Math. Statist. and Prob., vol. IV, pp. 223–239. University of California Press, Berkeley, Calif (1961)
Garet O., Marchand R.: Coexistence in two-type first-passage percolation models. Ann. Appl. Probab. 15(1A), 298–330 (2005)
Gouéré J.-B.: Shape of territories in some competing growth models. Ann. Appl. Probab. 17(4), 1273–1305 (2007)
Grimmett G., Marstrand J.M.: The supercritical phase of percolation is well behaved. Proc. R. Soc. Lond. Ser. A 430, 439–457 (1990)
Häggström O., Meester R.: Asymptotic shapes for stationary first passage percolation. Ann. Probab. 23, 1511–1522 (1995)
Häggström O., Pemantle R.: First passage percolation and a model for competing spatial growth. J. Appl. Probab. 35, 683–692 (1998)
Hammersley, J., Welsh, D.: First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In: Proc. Internat. Res. Semin., Statist. Lab., University of California, Berkeley, Calif, pp. 61–110. Springer-Verlag, New York (1965)
Hoffman C.: Geodesics in first passage percolation. Ann. Appl. Probab. 18, 1944–1969 (2008)
Hoffman C.: Coexistence for Richardson type competing spatial growth models. Ann. Appl. Probab. 15(1B), 739–747 (2005)
Howard, C. Douglas: Models of first-passage percolation. In: Probability on Discrete Structures. Encyclopaedia Math. Sci., vol. 110, pp. 125–173 Springer, Berlin (2004)
Howard C.D., Newman C.M.: Euclidean models of first-passage percolation. Probab. Theory Relat. Fields 108, 153–170 (1997)
Johansson K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000)
Kardar K., Parisi G., Zhang Y.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889–892 (1986)
Kesten H.: On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3, 296–338 (1993)
Kesten, H.: First-passage percolation. In: From Classical to Modern Probability. Progr. Probab., vol. 54, 93–143. Birkhäuser, Basel (2003)
Marchand R.: Strict inequalities for the time constant in first passage percolation. Ann. Appl. Probab. 12, 1001–1038 (2002)
Newman, C.: A surface view of first-passage percolation. In: Proceedings of the International Congress of Mathematicians, vols. 1, 2 (Zürich, 1994), pp. 1017–1023, Birkhäuser, Basel (1995)
Newman C., Piza M.: Divergence of shape fluctuations in two dimensions. Ann. Probab. 23, 977–1005 (1995)
Pemantle, R., Peres, Y.: Planar first-passage percolation times are not tight. In: Probability and Phase Transition (Cambridge, 1993). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 420, pp. 261–264. Kluwer Acad. Publ., Dordrecht (1994)
Pimentel L.P.R.: Multitype shape theorems for first passage percolation models. Adv. Appl. Probab. 39(1), 53–76 (2007)
Richardson D.: Random growth in a tessellation. Proc. Cambridge Philos. Soc. 74, 515–528 (1973)
Vanden Berg J., Kesten H.: Inequalities for the time constant in first-passage percolation. Ann. Appl. Probab. 3, 56–80 (1993)
Yaglom I., Boltjanski V.: Convex figures. Holt, Rinehart and Winston, New York (1960)
Zhang Yu.: Shape fluctuations are different in different directions. Ann. Probab. 36, 331–362 (2008)
Zhang Yu.: On the concentration and the convergence rate with a moment condition in first-passage percolation. Stoch. Proc. Appl. 120, 1317–1341 (2010)
Zhang Yu.: The time constant vanishes only on the percolation cone in directed first passage percolation. Electron. J. Prob. 77, 2264–2286 (2009)
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A. Auffinger’s research partially funded by NSF Grant DMS 0806180. M. Damron’s research funded by an NSF Postdoctoral Fellowship.
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Auffinger, A., Damron, M. Differentiability at the edge of the percolation cone and related results in first-passage percolation. Probab. Theory Relat. Fields 156, 193–227 (2013). https://doi.org/10.1007/s00440-012-0425-4
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DOI: https://doi.org/10.1007/s00440-012-0425-4
Keywords
- First-passage percolation
- Shape fluctuations
- Oriented percolation
- Richardson’s growth model
- Graph of infection