Electronic Journal of Probability

Asymptotic behavior of the Brownian frog model

Erin Beckman, Emily Dinan, Rick Durrett, Ran Huo, and Matthew Junge

Full-text: Open access

Abstract

We introduce an extension of the frog model to Euclidean space and prove properties for the spread of active particles. Fix $r>0$ and place a particle at each point $x$ of a unit intensity Poisson point process $\mathcal{P} \subseteq \mathbb{R} ^d - \mathbb{B} (0,r)$. Around each point in $\mathcal{P} $, put a ball of radius $r$. A particle at the origin performs Brownian motion. When it hits the ball around $x$ for some $x \in \mathcal{P} $, new particles begin independent Brownian motions from the centers of the balls in the cluster containing $x$. Subsequent visits to the cluster do nothing. This waking process continues indefinitely. For $r$ smaller than the critical threshold of continuum percolation, we show that the set of activated points in $\mathcal{P} $ approximates a linearly expanding ball. Moreover, in any fixed ball the set of active particles converges to a unit intensity Poisson point process.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 104, 19 pp.

Dates
Received: 15 December 2017
Accepted: 21 August 2018
First available in Project Euclid: 20 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1540000928

Digital Object Identifier
doi:10.1214/18-EJP215

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces

Keywords
frog model shape theorem continuum percolation

Rights
Creative Commons Attribution 4.0 International License.

Citation

Beckman, Erin; Dinan, Emily; Durrett, Rick; Huo, Ran; Junge, Matthew. Asymptotic behavior of the Brownian frog model. Electron. J. Probab. 23 (2018), paper no. 104, 19 pp. doi:10.1214/18-EJP215. https://projecteuclid.org/euclid.ejp/1540000928


Export citation

References

  • [AMP02a] O. S. M. Alves, F. P. Machado, and S. Yu. Popov. The shape theorem for the frog model. Ann. Appl. Probab., 12(2):533–546, 2002.
  • [AMP02b] Oswaldo Alves, Fabio Machado, and Serguei Popov. Phase transition for the frog model. Electron. J. Probab., 7:no. 16, 1–21, 2002.
  • [AMPR01] O. S. M. Alves, F. P. Machado, S. Yu. Popov, and K. Ravishankar. The shape theorem for the frog model with random initial configuration. Markov Process. Related Fields, 7(4):525–539, 2001.
  • [BG81] Maury Bramson and David Griffeath. On the Williams-Bjerknes tumour growth model i. The Annals of Probability, 9(2):173–185, 1981.
  • [DZ10] Amir Dembo and Ofer Zeitouni. Large Deviations Techniques and Applications, volume 38. Springer, 2010.
  • [DP14] Christian Döbler and Lorenz Pfeifroth. Recurrence for the frog model with drift on $\mathbb{Z} ^d$. Electron. Commun. Probab., 19:no. 79, 13, 2014.
  • [Dur10] Richard Durrett. Probability: Theory and Examples. Number v. 3 in Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 2010.
  • [DG82] Richard Durrett and David Griffeath. Contact processes in several dimensions. Probability Theory and Related Fields, 59(4):535–552, 1982.
  • [GS09] Nina Gantert and Philipp Schmidt. Recurrence for the frog model with drift on $\mathbb{Z} $. Markov Process. Related Fields, 15(1):51–58, 2009.
  • [Her18] J. Hermon. Frogs on trees ? Electron. J. Probab., 23:no. 17, 40 pp, 2018.
  • [HJJ16a] Christopher Hoffman, Tobias Johnson, and Matthew Junge. From transience to recurrence with Poisson tree frogs. Ann. Appl. Probab., 26(3):1620–1635, 2016.
  • [HJJ16b] Christopher Hoffman, Tobias Johnson, and Matthew Junge. Recurrence and transience for the frog model on trees. Ann. Probab., Volume 45, Number 5: 2826-2854,.
  • [HJJ17] Christopher Hoffman, Tobias Johnson, and Matthew Junge. Infection spread for the frog model on trees. arXiv:1710.05884, 2017.
  • [HJJ18] Christopher Hoffman, Tobias Johnson, and Matthew Junge. Cover time for the frog model on trees. arXiv:1802.03428, 2018.
  • [JJ16] Tobias Johnson and Matthew Junge. The critical density for the frog model is the degree of the tree. Electron. Commun. Probab. 21:no. 82, 12, 2016.
  • [Kub16] N. Kubota. Deviation bounds for the first passage time in the frog model. ArXiv e-prints, December 2016.
  • [LMP05] Élcio Lebensztayn, Fábio P. Machado, and Serguei Popov. An improved upper bound for the critical probability of the frog model on homogeneous trees. J. Stat. Phys., 119(1-2):331–345, 2005.
  • [MR96] Ronald Meester and Rahul Roy. Continuum percolation, volume 119. Cambridge University Press, 1996.
  • [Pen03] Mathew Penrose. Random geometric graphs. Number 5. Oxford University Press, 2003.
  • [Pop01] Serguei Yu. Popov. Frogs in random environment. J. Statist. Phys., 102(1-2):191–201, 2001.
  • [RS04] Alejandro F. Ramírez and Vladas Sidoravicius. Asymptotic behavior of a stochastic combustion growth process. J. Eur. Math. Soc. (JEMS), 6(3):293–334, 2004.
  • [Ros17a] J. Rosenberg. Recurrence of the frog model on the 3,2-alternating tree. ALEA, Lat. Am. J. Probab. Math. Stat. 15: 811–836, 2018.
  • [Ros17b] J. Rosenberg. The frog model with drift on $\mathbb R$. Electron. Commun. Probab., 22:no. 30, 14 pp, 2017.
  • [Roy90] Rahul Roy. The RSW theorem and the equality of critical densities and the ‘dual’ critical densities for continuum percolation on $\mathbb{R} ^2$. 18:1563–1575, 01 1990.
  • [TW99] András Telcs and Nicholas C. Wormald. Branching and tree indexed random walks on fractals. J. Appl. Probab., 36(4):999–1011, 1999.
  • [vdBBdH01] M. van den Berg, E. Bolthausen, and F. den Hollander. Moderate deviations for the volume of the Wiener sausage. Annals of Mathematics, 153(2):355–406, 2001.