Abstract
We survey recent rigorous results and open problems related to models of Interacting Particle Systems which describe the autocatalytic type reaction A+B→2B, with diffusion constants of particles being respectively D A ≥0 and D B ≥0. Depending on the choice of the values of D A and D B , we cover three distinct cases: the so called “rumor or infection spread” model (D A >0,D B >0); the Stochastic Combustion process (D A =0 and D B >0); and finally the “modified” Diffusion Limited Aggregation, which corresponds to the case D A >0, D B =0 with modified transition rule: A+B→2B occurs when an A- and a B-particles become nearest neighbors and the A-particle attempts to jump on a vertex where the B-particle is located. Then such jump is suppressed, and A-particle becomes B-particle.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alves, O., Machado, F., Popov, S.: The shape theorem for the frog model Ann. Appl. Probab. 12, 533–546 (2002)
Bérard, J., Ramírez, A.F.: Large deviations of the front in a one dimensional model of X + Y → 2X. In: Bérard, J., Ramírez, A.F. Ann. Probab. 38(3), 955–1018 (2010)
Bramson, M.: Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Amer. Math. Soc. 44(285), iv+190 (1983)
Bramson, M., Griffeath. D.: On the Williams-Bjerknes Tumour Growth Model II. Math. Proc. Cambridge Philos. Soc. 88, 339–357 (1980)
Comets, F., Quastel, J., Ramírez, A.F.: Fluctuations of the front in a stochastic combustion model, Ann. de l’IHP, Probabilités et Statistiques, Vol. 43. 147–162 (2007)
Comets, F., Quastel, J., Ramírez, A.F.: Fluctuations of the Front in a one dimensional model of X + Y → 2X. Trans. Amer. Math. Soc. 361, 6165–6189 (2009)
Chayes, L., Swindle, G.: Hydrodynamic limits for one-dimensional particle systems with moving boundaries. Ann. Probab. 24(2), 559–598 (1996)
Cox, J.T., Durrett, R.: The stepping stone model: new formulas expose old myths, Ann. Appl. Probab. 12, 1348–1377 (2002)
Eden, M.: A two dimensional growth process, in Fourth Berkeley sympos. Math. Statist. Probab. IV, 223–239 (1961); In: Neyman, J. (eds.) University of California Press, Berkeley, CA.
Dickman, R., Rolla, L.T., Sidoravicius, V.: Activated Random Walkers: Facts, Conjectures and Challanges, J. Stat. Physics. 138, 126–142 (2010)
Garet, O., Marchand, R.: Asymptotic shape for the chemical distance and first-passage percolation in random environment. ESAIM: Probab. Stat. 8, 169–199 (2004)
Gaudilliere, A., Nardi, F.: An upper bound for front propagation velocities inside moving populations. Braz. J. Probab. Stat. 24, 256–278 (2010)
Hammersley, J.M.: Postulates for subadditive processes. Ann. Probab. 2, 652–680 (1974)
Hammersley, J.M., Welsh, D.J.A.: First-passage percolation, subadditive processes, stochastic networks and generalized renewal theory, in Bernoulli, Bayes, Laplace Anniversary Volume. (J. Neyman and L.M. LeCam, eds.), pp. 61–110. Springer, New York (1965)
Howard, C.D.: Models of first passage percolation, in Probability on discrete structures (H. Kesten, eds.) pp. 125–173. Springer, New York (2003)
Jara, M., Moreno, G., Ramírez, A.F.: Front propagation in an exclusion one-dimensional reactive dynamics. Markov Process. Related Fields 18, 185–206 (2008)
Kesten, H., Sidoravicius, V.: Branching random walk with catalysts, Elec. J. Probab. 8, paper # 6 (2003)
Kesten, H.: Aspects of first passage percolation, in Lecture Notes in Mathematics. vol. 1180, pp. 125–264. Springer, New York (1986)
Kesten, H., Sidoravicius, V.: The spread of a rumor or infection in a moving population, Ann. Probab. 33(6), 2402–2462 (2005)
Kesten, H., Sidoravicius, V.: A shape theorem for the spread of an infection, Ann. Math. 167, 701–766 (2008)
Kesten, H., Sidoravicius, V.: A phase transition in a model for the spread of an infection. Illinois J. Math. 50(3), 547–634 (2006)
Kesten, H., Sidoravicius, V.: A problem in one-dimensional diffusion-limited aggregation (DLA) and positive recurrence of Markov chains, Ann. of Probab. 36(5), 1838–1879 (2008)
Kesten, H., Sidoravicius, V.: Positive recurrence of a one-dimensional variant of diffusion limited aggregation. In and out of equilibrium. vol. 2, pp. 429–461. Progr. Probab., 60, Birkhäuser, Basel (2008)
Kingman, J.F.C.: Subadditive processes, in Lecture Notes in Mathematics. vol. 539, pp. 168–223. Springer, New York (1975)
Lawler, G.F., Bramson, M., Griffeath, D.: Internal Diffusion Limited Aggregation. Ann. Probab. 20(4), 2117–2140 (1992)
Panja, D.: Effects of fluctuations on propagating fronts, Physics Reports. 393, 87–174 (2004)
Ramírez, A.F., Sidoravicius, V.: Asymptotic behavior of a stochastic combustion growth process, J. Eur. Math. Soc. 6(3), 293–334 (2004)
Richardson, D.: Random growth in a tesselation, Proc. Cambridge Philos. Soc. 74, 515–528 (1973)
Rolla, L., Sidoravicius, V.: Absorbing-state phase transition for stochastic sandpiles and activated random walks. Arxiv:09081152
Sznitman, A.S., Zerner, M.: A law of large numbers for random walks in random environment, Ann. Probab. 27(4), 1851–1869 (1999)
Voss, R.F.: Multiparticle fractal aggregation. J. Stat. Phys. 36(5–6), 861–872 (1984)
Wierman, J.C.: The front velocity of the simple epidemic. J. Appl. Probab. 16, 409–415 (1979)
Witten, Jr. T.A., Sander, L.M.: Diffusion-Limited aggregation, a kinetic critical phenomenon. Phys. Rev. Lett. 47, 1400–1403 (1981)
Acknowledgements
A.F.R and V.S. would like to thank the all organizers of both workshops and in particular to Prof. Wolfgang König for his hospitality. A.F.R. would like to thank the support of FONDECYT grant 1100298.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kesten, H., Ramı́rez, A.F., Sidoravicius, V. (2012). Asymptotic Shape and Propagation of Fronts for Growth Models in Dynamic Random Environment. In: Deuschel, JD., Gentz, B., König, W., von Renesse, M., Scheutzow, M., Schmock, U. (eds) Probability in Complex Physical Systems. Springer Proceedings in Mathematics, vol 11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23811-6_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-23811-6_8
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-23810-9
Online ISBN: 978-3-642-23811-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)