Electronic Journal of Probability

Localization for a Class of Linear Systems

Yukio Nagahata and Nobuo Yoshida

Full-text: Open access

Abstract

We consider a class of continuous-time stochastic growth models on d-dimensional lattice with non-negative real numbers as possible values per site. The class contains examples such as binary contact path process and potlatch process. We show the equivalence between the slow population growth and localization property that the time integral of the replica overlap diverges. We also prove, under reasonable assumptions, a localization property in a stronger form that the spatial distribution of the population does not decay uniformly in space.

Article information

Source
Electron. J. Probab., Volume 15 (2010), paper no. 20, 636-653.

Dates
Accepted: 17 May 2010
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819805

Digital Object Identifier
doi:10.1214/EJP.v15-757

Mathematical Reviews number (MathSciNet)
MR2650776

Zentralblatt MATH identifier
1226.60134

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J75: Jump processes

Keywords
localization linear systems binary contact path process potlatch process

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Nagahata, Yukio; Yoshida, Nobuo. Localization for a Class of Linear Systems. Electron. J. Probab. 15 (2010), paper no. 20, 636--653. doi:10.1214/EJP.v15-757. https://projecteuclid.org/euclid.ejp/1464819805


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References

  • Carmona, Philippe; Hu, Yueyun. On the partition function of a directed polymer in a Gaussian random environment. Probab. Theory Related Fields 124 (2002), no. 3, 431–457.
  • Carmona, Philippe; Hu, Yueyun. Strong disorder implies strong localization for directed polymers in a random environment. ALEA Lat. Am. J. Probab. Math. Stat. 2 (2006), 217–229.
  • Comets, Francis; Shiga, Tokuzo; Yoshida, Nobuo. Directed polymers in a random environment: path localization and strong disorder. Bernoulli 9 (2003), no. 4, 705–723.
  • Comets, Francis; Yoshida, Nobuo. Brownian directed polymers in random environment. Comm. Math. Phys. 254 (2005), no. 2, 257–287.
  • Comets, Francis; Yoshida, Nobuo. Branching Random Walks in Space-Time Random Environment: Survival Probability, Global and Local Growth Rates, preprint (2009), arXiv:0907.0509, to appear in J. Theoret. Prob.
  • Griffeath, David. The binary contact path process. Ann. Probab. 11 (1983), no. 3, 692–705.
  • He, Sheng Wu; Wang, Jia Gang; Yan, Jia An. Semimartingale theory and stochastic calculus. Kexue Chubanshe (Science Press), Beijing; CRC Press, Boca Raton, FL, 1992. xiv+546 pp. ISBN: 7-03-003066-4
  • Holley, Richard; Liggett, Thomas M. Generalized potlatch and smoothing processes. Z. Wahrsch. Verw. Gebiete 55 (1981), no. 2, 165–195.
  • Hu, Yueyun; Yoshida, Nobuo. Localization for branching random walks in random environment. Stochastic Process. Appl. 119 (2009), no. 5, 1632–1651.
  • Liggett, Thomas M. Interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 276. Springer-Verlag, New York, 1985. xv+488 pp. ISBN: 0-387-96069-4
  • Liggett, Thomas M.; Spitzer, Frank. Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrsch. Verw. Gebiete 56 (1981), no. 4, 443–468.
  • Nagahata, Yukio; Yoshida, Nobuo. Central limit theorem for a class of linear systems. Electron. J. Probab. 14 (2009), no. 34, 960–977. (Review)
  • Nagahata, Yukio; Yoshida, Nobuo. A Note on the Diffusive Scaling Limit for a Class of Linear Systems. Electron. Comm. Probab. 15 (2010), no. 7, 68–78.
  • Shiozawa, Yuichi. Localization for branching Brownian motions in random environment. Tohoku Math. J. 61 (2009), no. 4, 483–497.
  • Spitzer, Frank. Infinite systems with locally interacting components. Ann. Probab. 9 (1981), no. 3, 349–364.
  • Yoshida, Nobuo. Phase transitions for the growth rate of linear stochastic evolutions. J. Stat. Phys. 133 (2008), no. 6, 1033–1058.
  • Yoshida, Nobuo. Localization for Linear Stochastic Evolutions J. Stat. Phys. 138 (2010), no. 4/5, 568–618.