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

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- by Francis Comets, Jeremy Quastel and Alejandro F. Ramírez PDF
- Trans. Amer. Math. Soc.
**361**(2009), 6165-6189 Request permission

## 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

- O. S. M. Alves, F. P. Machado, and S. Yu. Popov,
*The shape theorem for the frog model*, Ann. Appl. Probab.**12**(2002), no. 2, 533–546. MR**1910638**, DOI 10.1214/aoap/1026915614 - 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**(2001), no. 4, 525–539. MR**1893139** - Martin T. Barlow, Robin Pemantle, and Edwin A. Perkins,
*Diffusion-limited aggregation on a tree*, Probab. Theory Related Fields**107**(1997), no. 1, 1–60. MR**1427716**, DOI 10.1007/s004400050076 - Antar Bandyopadhyay and Ofer Zeitouni,
*Random walk in dynamic Markovian random environment*, ALEA Lat. Am. J. Probab. Math. Stat.**1**(2006), 205–224. MR**2249655** - M. Bramson, P. Calderoni, A. De Masi, P. Ferrari, J. Lebowitz, and R. H. Schonmann,
*Microscopic selection principle for a diffusion-reaction equation*, J. Statist. Phys.**45**(1986), no. 5-6, 905–920. MR**881315**, DOI 10.1007/BF01020581 - Francis Comets, Jeremy Quastel, and Alejandro F. Ramírez,
*Fluctuations of the front in a stochastic combustion model*, Ann. Inst. H. Poincaré Probab. Statist.**43**(2007), no. 2, 147–162 (English, with English and French summaries). MR**2303116**, DOI 10.1016/j.anihpb.2006.01.005 - Harry Kesten,
*A renewal theorem for random walk in a random environment*, Probability (Proc. Sympos. Pure Math., Vol. XXXI, Univ. Illinois, Urbana, Ill., 1976) Amer. Math. Soc., Providence, R.I., 1977, pp. 67–77. MR**0458648** - Harry Kesten and Vladas Sidoravicius,
*The spread of a rumor or infection in a moving population*, Ann. Probab.**33**(2005), no. 6, 2402–2462. MR**2184100**, DOI 10.1214/009117905000000413 - John Lamperti,
*Stochastic processes*, Applied Mathematical Sciences, Vol. 23, Springer-Verlag, New York-Heidelberg, 1977. A survey of the mathematical theory. MR**0461600** - 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. - Panja, D. (2004).
*Effects of Fluctuations on Propagating Fronts*, Physics Reports**393**, 87-174. - V. V. Petrov,
*Sums of independent random variables*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by A. A. Brown. MR**0388499** - A. F. Ramírez and V. Sidoravicius,
*Asymptotic behavior of a stochastic combustion growth process*, J. Eur. Math. Soc. (JEMS)**6**(2004), no. 3, 293–334. MR**2060478** - van Saarloos, W. (2003).
*Front propagation into unstable states*, Phys. Rep.**386**, 29. - Alain-Sol Sznitman,
*Slowdown estimates and central limit theorem for random walks in random environment*, J. Eur. Math. Soc. (JEMS)**2**(2000), no. 2, 93–143. MR**1763302**, DOI 10.1007/s100970050001 - Alain-Sol Sznitman and Martin Zerner,
*A law of large numbers for random walks in random environment*, Ann. Probab.**27**(1999), no. 4, 1851–1869. MR**1742891**, DOI 10.1214/aop/1022874818 - Hermann Thorisson,
*Coupling, stationarity, and regeneration*, Probability and its Applications (New York), Springer-Verlag, New York, 2000. MR**1741181**, DOI 10.1007/978-1-4612-1236-2

## 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
- MR Author ID: 322635
- 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
- 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 - © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2529928