Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Fluctuations of the front in a one dimensional model of $ X+Y\to 2X$


Authors: Francis Comets, Jeremy Quastel and Alejandro F. Ramírez
Journal: Trans. Amer. Math. Soc. 361 (2009), 6165-6189
MSC (2000): Primary 82C22, 82C41; Secondary 82C24, 60K05, 60G50
DOI: https://doi.org/10.1090/S0002-9947-09-04889-2
Published electronically: May 1, 2009
MathSciNet review: 2529928
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a model of the reaction $ X+Y\to 2X$ on the integer lattice in which $ Y$ particles do not move while $ X$ particles move as independent continuous time, simple symmetric random walks. $ Y$ particles are transformed instantaneously to $ X$ particles upon contact. We start with a fixed number $ a\ge 1$ of $ Y$ particles at each site to the right of the origin. We prove a central limit theorem for the rightmost visited site of the $ X$ particles up to time $ t$ and show that the law of the environment as seen from the front converges to a unique invariant measure.


References [Enhancements On Off] (What's this?)

  • 1. Alves, O.; Machado, F.; Popov, S. (2002). The shape theorem for the frog model, Ann. Appl. Probab., 12, no. 2, 533-546. MR 1910638 (2003c:60159)
  • 2. Alves, O.; Machado, F.; Popov, S.; Ravishankar, K. (2001). The shape theorem for the frog model with random initial configuration, Markov Processes Relat. Fields 7 (4), 525-539. MR 1893139 (2003f:60171)
  • 3. Barlow, M.; Pemantle, R.; Perkins, E. (1997) Diffusion limited aggregation on a tree, Probab. Theory and Related Fields 97, no. 1, 1-60. MR 1427716 (97m:60146)
  • 4. Bandyopadhyay, A.; Zeitouni, O. (2006) Random Walk in Dynamic Markovian Random Environment, ALEA Lat. Am. J. Probab. Math. Stat. 1, 205-224 MR 2249655 (2007e:60097)
  • 5. Bramson, M.; Calderoni, P.; De Masi, A.; Ferrari, P.; Lebowitz, J.; Schonmann, R. (1986). Microscopic selection principle for a diffusion-reaction equation, J. Statist. Phys. 45, no. 5-6, 905-920. MR 881315 (88h:60198)
  • 6. Comets, F.; Quastel, J.; Ramírez, A.F. (2007) Fluctuations of the front in a stochastic combustion model, Ann. Inst. H. Poincaré Probab. Statist. 43(2), 147-162. MR 2303116 (2008b:60211)
  • 7. Kesten, H. (1977). A renewal theorem for random walk in a random environment, Proc. Symp. Pure Math. 31, 67-77. MR 0458648 (56:16848)
  • 8. Kesten, H.; Sidoravicius, V. (2005). The spread of a rumor or infection in a moving population, Ann. Probab. 33, no. 6, 2402-2462. MR 2184100 (2006m:60139)
  • 9. Lamperti, J. (1977). Stochastic Processes: A survey of the mathematical theory. Springer-Verlag. MR 0461600 (57:1585)
  • 10. Mai, J.; Sokolov, I.M.; Kuzovkov, V.N.; Blumen, A. (1997). Front form and velocity in a one-dimensional auto-catalytic A+B$ \to$2A reaction, Phys. Rev. E 56, 4130-4134.
  • 11. Panja, D. (2004). Effects of Fluctuations on Propagating Fronts, Physics Reports 393, 87-174.
  • 12. Petrov, V. (1975). Sums of independent random variables, Springer-Verlag. MR 0388499 (52:9335)
  • 13. Ramírez, A. F.; Sidoravicius, V. (2004). Asymptotic behavior of a stochastic combustion growth process, J. Eur. Math. Soc. 6, no. 3, 293-334. MR 2060478 (2005e:60234)
  • 14. van Saarloos, W. (2003). Front propagation into unstable states, Phys. Rep. 386, 29.
  • 15. Sznitman, A.S. (2000). Slowdown estimates and central limit theorem for random walks in random environment, J. Eur. Math. Soc. 2, no. 2, 93-143. MR 1763302 (2001j:60192)
  • 16. Sznitman, A.S.; Zerner, M. (1999). A law of large numbers for random walks in random environment, Ann. Probab., 27, 4, 1851-1869. MR 1742891 (2001f:60116)
  • 17. Thorisson, H.(2000). Coupling, stationarity, and regeneration, Probability and its Applications (New York). Springer-Verlag, New York. MR 1741181 (2001b:60003)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 82C22, 82C41, 82C24, 60K05, 60G50

Retrieve articles in all journals with MSC (2000): 82C22, 82C41, 82C24, 60K05, 60G50


Additional Information

Francis Comets
Affiliation: Laboratoire de Probabilités et Modèles Aléatoires, Université Paris 7- Denis Diderot, 2, Place Jussieu, F-75251 Paris Cedex 05, France
Email: comets@math.jussieu.fr

Jeremy Quastel
Affiliation: Departments of Mathematics and Statistics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada M5S 1L2
Email: quastel@math.toronto.edu

Alejandro F. Ramírez
Affiliation: Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Macul, Santiago, Chile
Email: aramirez@mat.puc.cl

DOI: https://doi.org/10.1090/S0002-9947-09-04889-2
Keywords: Regeneration times, interacting particle systems, front propagation
Received by editor(s): April 26, 2007
Received by editor(s) in revised form: July 30, 2008
Published electronically: May 1, 2009
Additional Notes: The first author was partially supported by CNRS, UMR 7599 and by ECOS-Conicyt grant CO5EO2
The second author was partially supported by NSERC, Canada
The third author was partially supported by Fondo Nacional de Desarrollo Científico y Tecnológico grant 1060738, by Iniciativa Científica Milenio P04-069-F, and by ECOS-Conicyt grant CO5EO2
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society