Central Limit Theorem for a Class of Linear Systems

Yukio Nagahata (Osaka University)
Nobuo Yoshida (Kyoto University)

Abstract


We consider a class of interacting particle systems with values in $[0,∞)^{\mathbb{Z}^d}$, of which the binary contact path process is an example. For $d \geq 3$ and under a certain square integrability condition on the total number of the particles, we prove a central limit theorem for the density of the particles, together with upper bounds for the density of the most populated site and the replica overlap.

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Pages: 960-977

Publication Date: May 5, 2009

DOI: 10.1214/EJP.v14-644

References

  1. Comets, F., Yoshida, N.: Some New Results on Brownian Directed Polymers in Random Environment, RIMS Kokyuroku {\bf 1386}, 50--66, available at authors' web pages.
  2. Chung, Kai Lai; Zhao, Zhong Xin. From Brownian motion to Schrödinger's equation.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 312. Springer-Verlag, Berlin, 1995. xii+287 pp. ISBN: 3-540-57030-6 MR1329992 (96f:60140)
  3. Davies, E. B. Heat kernels and spectral theory.Cambridge Tracts in Mathematics, 92. Cambridge University Press, Cambridge, 1989. x+197 pp. ISBN: 0-521-36136-2 MR0990239 (90e:35123)
  4. Griffeath, David. The binary contact path process. Ann. Probab. 11 (1983), no. 3, 692--705. MR0704556 (85b:60097)
  5. 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 MR0776231 (86e:60089)
  6. Nakashima, M.: The Central Limit Theorem for Linear Stochastic Evolutions, preprint (2008), to appear in J. Math. Kyoto Univ.
  7. Spitzer, Frank. Principles of random walks.Second edition.Graduate Texts in Mathematics, Vol. 34.Springer-Verlag, New York-Heidelberg, 1976. xiii+408 pp. MR0388547 (52 #9383)
  8. Sznitman, Alain-Sol. Brownian motion, obstacles and random media.Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. xvi+353 pp. ISBN: 3-540-64554-3 MR1717054 (2001h:60147)
  9. Woess, Wolfgang. Random walks on infinite graphs and groups.Cambridge Tracts in Mathematics, 138. Cambridge University Press, Cambridge, 2000. xii+334 pp. ISBN: 0-521-55292-3 MR1743100 (2001k:60006)
  10. Yoshida, Nobuo. Central limit theorem for branching random walks in random environment. Ann. Appl. Probab. 18 (2008), no. 4, 1619--1635. MR2434183 (Review)
  11. Yoshida, Nobuo. Phase transitions for the growth rate of linear stochastic evolutions. J. Stat. Phys. 133 (2008), no. 6, 1033--1058. MR2462010
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