Electronic Journal of Probability

First passage time of the frog model has a sublinear variance

Van Hao Can and Shuta Nakajima

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In this paper, we show that the first passage time in the frog model on $\mathbb{Z} ^{d}$ with $d\geq 2$ has a sublinear variance. This implies that the central limit theorem does not hold at least with the standard diffusive scaling. The proof is based on the method introduced in [4, 11] combined with a control of the maximal weight of paths in a locally dependent site-percolation. We also apply this method to get the linearity of the lengths of optimal paths.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 76, 27 pp.

Received: 1 October 2018
Accepted: 15 June 2019
First available in Project Euclid: 5 July 2019

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

frog model first passage time sublinear variance

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Can, Van Hao; Nakajima, Shuta. First passage time of the frog model has a sublinear variance. Electron. J. Probab. 24 (2019), paper no. 76, 27 pp. doi:10.1214/19-EJP334. https://projecteuclid.org/euclid.ejp/1562292237

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  • [1] O. S. M. Alves, F. P. Machado, S. P. Popov. The shape theorem for the frog model, Ann. Appl. Probab. 12, 533–546, (2002).
  • [2] O. S. M. Alves, F. P. Machado, S. P. Popov. Phase transition for the frog model. Electron. J. Probab. 7, no. 16, 21 pp. (2002).
  • [3] E. Beckman, E. Dinan, R. Durrett, R. Huo, M. Junge. Asymptotic behavior of the Brownian frog model. Electron. J. Probab. 23, no. 104, 19 pp. (2018).
  • [4] M. Benaïm, R. Rossignol. Exponential concentration for first passage percoaltion through modified Poincaré inequalities, Ann. Inst. Henri Poincaré. Prob. Stat. 44, 544–573 (2008).
  • [5] I. Benjamini, G. Kalai, O. Schramm. First-passage percolation has sublinear distance variance, Ann. Probab. 31, 1970–1978, (2003).
  • [6] J. Bérard, A. F. Ramírez. Large deviations of the front in a one-dimensional model of $X+Y \rightarrow 2X$. Ann. Probab. 38, 955–1018, (2010).
  • [7] S. Boucheron, G. Lugosi, P. Massart. Concentration inequalities. Oxford University Press, Oxford, (2013).
  • [8] V. H. Can, S. Nakajima, N. Kubota. Large deviations of the first passage time in the frog model, in preparation.
  • [9] Sourav Chatterjee. Superconcentration and Related Topics, Springer Monographs in Mathematics, (2014).
  • [10] F. Comets, J. Quastel, A. F. Ramírez. Fluctuations of the front in a one dimensional model of $X +Y \rightarrow 2X$. Trans. Amer. Math. Soc. 361, 6165–6189 (2009).
  • [11] M. Damron, J. Hanson, P. Sosoe. Sublinear variance in first-passage percolation for general distributions, Probab. Theo. Rel. Fields. 163, 223–258, (2015).
  • [12] C. Döbler, N. Gantert, T. Höfelsauer, S. Popov, F. Weidner. Recurrence and transience of frogs with drift on $\mathbb{Z} ^{d}$. Electron. J. Probab. 23, no. 88, 23 pp. (2018).
  • [13] C. Döbler, L. Pfeifroth. Recurrence for the frog model with drift on $Z^{d}$. Electron. Commun. Probab. 19, 13 pp, (2014).
  • [14] N. Gantert, P. Schmidt. Recurrence for the frog model with drift on $\mathbb{Z} $. Markov Process. Rel. Fields, 15, 51–58, (2009).
  • [15] C. Hoffman, T. Johnson, M. Junge. From transience to recurrence with Poisson tree frogs. Ann. Appl. Probab. 26, 1620–1635, (2016).
  • [16] C. Hoffman, T. Johnson, M. Junge. Recurrence and transience for the frog model on trees. Ann. Probab. 45, 2826–2854, (2017).
  • [17] J. F. C. Kingman. Subadditive ergodic theorem, Ann. Probab. 1, 883–909, (1973).
  • [18] E. Kosygina, M. P. W. Zerner. A zero–one law for recurrence and transience of frog processes. Probab. Theo. Rel. Fields, 168 317–346, (2017).
  • [19] N. Kubota. Deviation bounds for the first passage time in the frog model, to appear in Advances in Applied Probability, arXiv:1612.08581 (2016).
  • [20] G. F. Lawler, V. Limic. Random walk: a modern introduction, volume 123 of Cambridge Studies in Advanced Mathematics. Cambridge University Press (2010).
  • [21] G. Menz, A. Schlichting. Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape, Ann. Probab. 42, 1809–1884 (2014).
  • [22] S. Y. Popov. Frogs in random environment. J. Stat. Phys. 102, 191–201, (2001).
  • [23] A. Telcs, N. C. Wormald. Branching and tree indexed random walks on fractals. J. Appl. Probab. 36 999–1011, (1999).