Journal of Statistical Physics

, Volume 150, Issue 2, pp 285–298 | Cite as

Localization for a Random Walk in Slowly Decreasing Random Potential

  • Christophe Gallesco
  • Serguei Popov
  • Gunter M. Schütz


We consider a continuous time random walk X in a random environment on ℤ+ such that its potential can be approximated by the function V:ℝ+→ℝ given by \(V(x)=\sigma W(x) -\frac {b}{1-\alpha}x^{1-\alpha}\) where σW a Brownian motion with diffusion coefficient σ>0 and parameters b, α are such that b>0 and 0<α<1/2. We show that P-a.s. (where P is the averaged law) \(\lim_{t\to\infty} \frac{X_{t}}{(C^{*}(\ln\ln t)^{-1}\ln t)^{\frac{1}{\alpha}}}=1\) with \(C^{*}=\frac{2\alpha b}{\sigma^{2}(1-2\alpha)}\). In fact, we prove that by showing that there is a trap located around \((C^{*}(\ln\ln t)^{-1}\ln t)^{\frac{1}{\alpha}}\) (with corrections of smaller order) where the particle typically stays up to time t. This is in sharp contrast to what happens in the “pure” Sinai’s regime, where the location of this trap is random on the scale ln2 t.


KMT strong coupling Brownian motion with drift Localization Random walk in random environment Reversibility 



C.G. is grateful to FAPESP (grant 2009/51139-3) for financial support. G.M.S. thanks FAPESP (grant 2011/21089-4) and S.P. thanks CNPq (grant 301644/2011-0) for financial supports. C.G. and S.P. thank FAPESP (grant 2009/52379-8) for financial support. C.G. and G.M.S. thank NUMEC for kind hospitality.


  1. 1.
    Ambjörnsson, T., Banik, S.K., Krichevsky, O., Metzler, R.: Sequence sensitivity of breathing dynamics in heteropolymer DNA. Phys. Rev. Lett. 97, 128105 (2006) ADSCrossRefGoogle Scholar
  2. 2.
    Armendáriz, I., Loulakis, M.: Thermodynamic limit for the invariant measures in supercritical zero range processes. Probab. Theory Relat. Fields 145(1–2), 175–188 (2009) MATHCrossRefGoogle Scholar
  3. 3.
    Comets, F., Popov, S.: Limit law for transition probabilities and moderate deviations for Sinai’s random walk in random environment. Probab. Theory Relat. Fields 126(4), 571–609 (2003) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Ferrari, P., Landim, C., Sisko, V.: Condensation for a fixed number of independent random variables. J. Stat. Phys. 128(5), 1153–1158 (2007) MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    Fribergh, A., Gantert, N., Popov, S.: On slowdown and speedup of transient random walks in random environment. Probab. Theory Relat. Fields 147(1–2), 43–88 (2010) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Gallesco, C.: Meeting time of independent random walks in random environment. ESAIM, Probab. Stat. (2011). doi: 10.1051/ps/2011159 MathSciNetGoogle Scholar
  7. 7.
    Giacomin, G., Toninelli, F.L.: Smoothing of depinning transitions for directed polymers with quenched disorder. Phys. Rev. Lett. 96, 070602 (2006) ADSCrossRefGoogle Scholar
  8. 8.
    Giacomin, G., Toninelli, F.L.: On the irrelevant disorder regime of pinning models. Ann. Probab. 37(5), 1841–1875 (2009) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Großkinsky, S., Schütz, G.M., Spohn, H.: Condensation in the zero range process: stationary and dynamical properties. J. Stat. Phys. 113, 389–410 (2003) MATHCrossRefGoogle Scholar
  10. 10.
    Großkinsky, S., Chleboun, P., Schütz, G.M.: Instability of condensation in the zero-range process with random interaction. Phys. Rev. E 78, 030101(R) (2008) ADSCrossRefGoogle Scholar
  11. 11.
    Jeon, I., March, P., Pittel, B.: Size of the largest cluster under zero-range invariant measures. Ann. Probab. 28, 1162–1194 (2000) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Kafri, Y., Mukamel, D., Peliti, L.: Melting and unzipping of DNA. Eur. Phys. J. B 27, 135–146 (2002) ADSGoogle Scholar
  13. 13.
    Komlós, J., Major, P., Tusnády, G.: An approximation of partial sums of independent RV’s and the sample DF. II. Z. Wahrscheinlichkeitstheor. Verw. Geb. 34(1), 33–58 (1976) MATHCrossRefGoogle Scholar
  14. 14.
    Magdon-Ismail, M., Atiya, A., Pratap, A., Abu-Mostafa, Y.: On the maximum drawdown of a Brownian motion. J. Appl. Probab. 41, 147–161 (2004) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Menshikov, M.V., Wade, A.R.: Random walk in random environment with asymptotically zero perturbation. J. Eur. Math. Soc. 8(3), 491–513 (2006) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Menshikov, M.V., Wade, A.R.: Logarithmic speeds for one-dimensional perturbed random walk in random environment. Stoch. Process. Appl. 118(3), 389–416 (2008) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Mörters, P., Peres, Y.: Brownian Motion. Cambridge University Press, Cambridge (2010) MATHGoogle Scholar
  18. 18.
    Salminen, P., Vallois, P.: On maximum increase and decrease of Brownian motion. Ann. Inst. Henri Poincaré 43, 655–676 (2007) ADSMATHCrossRefGoogle Scholar
  19. 19.
    Sinai, Ya.G.: The limiting behavior of one-dimensional random walk in random medium. Theory Probab. Appl. 27, 256–268 (1982) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Zeitouni, O.: Random walks in random environment. In: Lectures on Probability Theory and Statistics. Lecture Notes in Math., vol. 1837, pp. 189–312. Springer, Berlin (2004) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Christophe Gallesco
    • 1
  • Serguei Popov
    • 1
  • Gunter M. Schütz
    • 2
  1. 1.Department of Statistics, Institute of Mathematics, Statistics and Scientific ComputationUniversity of Campinas–UNICAMPCampinasBrazil
  2. 2.Theoretical Soft Matter and Biophysics, Institute of Complex SystemsForschungszentrum JülichJülichGermany

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