Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-23T23:07:59.782Z Has data issue: false hasContentIssue false
  • English
  • Français

On the range of the transient frog model on ℤ

Part of: Basic hypergeometric functions Special processes Markov processes Distribution theory - Probability

Published online by Cambridge University Press:  26 June 2017

Arka Ghosh ,
Steven Noren  and
Alexander Roitershtein
Arka Ghosh*
Affiliation:
Iowa State University
Steven Noren*
Affiliation:
Iowa State University
Alexander Roitershtein*
Affiliation:
Iowa State University
*
* Postal address: Department of Statistics, Iowa State University, Ames, IA 50011, USA.
** Postal address: Department of Mathematics, Iowa State University, Ames, IA 50011, USA.
** Postal address: Department of Mathematics, Iowa State University, Ames, IA 50011, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We observe the frog model, an infinite system of interacting random walks, on ℤ with an asymmetric underlying random walk. For certain initial frog distributions we construct an explicit formula for the moments of the leftmost visited site, as well as their asymptotic scaling limits as the drift of the underlying random walk vanishes. We also provide conditions in which the lower bound can be scaled to converge in probability to the degenerate distribution at 1 as the drift vanishes.

Keywords

Frog modelinteracting random walkrandom walks rangeq-series

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60E05: Distributions 33D05: $q$-gamma functions, $q$-beta functions and integrals
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 2 , June 2017 , pp. 327 - 343
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Alves, O. S. M., Machado, F. P. and Popov, S. Y. (2002). The shape theorem for the frog model. Ann. Appl. Prob. 12, 533–546. CrossRefGoogle Scholar
[2] Alves, O. S. M., Machado, F. P. and Popov, S. Y. (2002). Phase transition for the frog model. Electron. J. Prob. 7, 21pp. CrossRefGoogle Scholar
[3] Alves, O. S. M., Machado, F. P., Popov, S. Y. and Ravishankar, K. (2001). The shape theorem for the frog model with random initial configuration. Markov Process. Relat. Fields 7, 525–539. Google Scholar
[4] Andrews, G. E. (1986). q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra (CBMS Reg. Conf. Ser. Math. 66). American Mathematical Society, Providence, RI. Google Scholar
[5] Bertacchi, D., Machado, F. P. and Zucca, F. (2014). Local and global survival for nonhomogeneous random walk systems on Z . Adv. Appl. Prob. 46, 256–278. CrossRefGoogle Scholar
[6] Charalambides, C. A. (2002). Enumerative Combinatorics. Chapman and Hall/CRC, Boca Raton, FL. Google Scholar
[7] Comets, F., Quastel, J. and Ramirez, A. F. (2007). Fluctuations of the front in a stochastic combustion model. Ann. Inst. H. Poincaré Prob. Statist. 43, 147–162. CrossRefGoogle Scholar
[8] Döbler, C. and Pfeifroth, L. (2014). Recurrence for the frog model with drift on Z d . Electron. Commun. Prob. 19, 13pp. CrossRefGoogle Scholar
[9] Durrett, R. (2010). Probability: Theory and Examples (Camb. Ser. Statist. Prob. Math. 31), 4th edn. Cambridge University Press. CrossRefGoogle Scholar
[10] Ernst, T. (2012). A Comprehensive Treatment of q-Calculus. Springer, Basel. CrossRefGoogle Scholar
[11] Eschenfeldt, P., Gross, B. and Pippenger, N. (2011). Asymptotic behavior of the moments of the maximum queue length during a busy period. Preprint. Available at https://arxiv.org/abs/1106.3590v1. Google Scholar
[12] Gantert, N. and Schmidt, P. (2009). Recurrence for the frog model with drift on Z . Markov Process. Relat. Fields 15, 51–58. Google Scholar
[13] Gaspar, G. and Rahman, M. (2004). Basic Hypergeometric Series (Encyclopedia Math. Appl. 96), 2nd edn. Cambridge University Press. CrossRefGoogle Scholar
[14] Gordon, B. and McIntosh, R. J. (2000). Some eighth order mock theta functions. J. London Math. Soc. 62, 321–335. CrossRefGoogle Scholar
[15] Hoffman, C., Johnson, T. and Junge, M. (2016). Recurrence and transience for the frog model on trees. Preprint. Available at https://arxiv.org/abs/1404.6238v6. Google Scholar
[16] Hoffman, C., Johnson, T. and Junge, M. (2016). From transience to recurrence with Poisson tree frogs. Ann. Appl. Prob. 26, 1620–1635. CrossRefGoogle Scholar
[17] Johnson, W. P. (2002). The curious history of Faà di Bruno's formula. Amer. Math. Monthly 109, 217–234. Google Scholar
[18] Lebensztayn, E., Machado, F. P. and Martinez, M. Z. (2008). Random walks systems with killing on Z . Stochastics 80, 451–457. CrossRefGoogle Scholar
[19] Lebensztayn, E., Machado, F. P. and Martinez, M. Z. (2010). Nonhomogeneous random walks systems on Z.. J. Appl. Prob. 47, 562–571. CrossRefGoogle Scholar
[20] Lukacs, E. (1955). Applications of Faà di Bruno's formula in mathematical statistics. Amer. Math. Monthly 62, 340–348. Google Scholar
[21] Peccati, G. and Taqqu, M. S. (2011). Wiener Chaos: Moments, Cumulants and Diagrams (Bocconi & Springer Ser. 1). Springer, Milan. CrossRefGoogle Scholar
[22] Popov, S. Y. (2001). Frogs in random environment. J. Statist. Phys. 102, 191–201. CrossRefGoogle Scholar
[23] Popov, S. Y. (2003). Frogs and some other interacting random walks models. In Discrete Random Walks (Discrete Math. Theoret. Comput. Sci. Proc. AC), Association of Discrete Mathematics and Theoretical Computer Science, Nancy, pp. 277–288. Google Scholar
[24] Ramirez, A. F. and Sidoravicius, V. (2004). Asymptotic behavior of a stochastic combustion growth process. J. Europ. Math. Soc. 6, 293–334. CrossRefGoogle Scholar
[25] Rudin, W. (1976). Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York. Google Scholar
[26] Telcs, A. and Wormald, N. C. (1999). Branching and tree-indexed random walks on fractals. J. Appl. Prob. 36, 999–1011. CrossRefGoogle Scholar