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Lyapunov exponents of Green’s functions for random potentials tending to zero


We consider quenched and annealed Lyapunov exponents for the Green’s function of −Δ +  γV, where the potentials \({V(x),\ x\in\mathbb {Z}^d}\), are i.i.d.  nonnegative random variables and γ > 0 is a scalar. We present a probabilistic proof that both Lyapunov exponents scale like \({c\sqrt{\gamma}}\) as γ tends to 0. Here the constant c is the same for the quenched as for the annealed exponent and is computed explicitly. This improves results obtained previously by Wang. We also consider other ways to send the potential to zero than multiplying it by a small number.

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  1. 1

    Durrett R.: Probability: Theory and Examples, 3rd edn. Duxbury Press, Belmont (2005)

  2. 2

    Flury M.: Large deviations and phase transition for random walks in random nonnegative potentials. Stoch. Proc. Appl. 117, 596–612 (2007)

  3. 3

    Flury M.: Coincidence of Lyapunov exponents for random walks in weak random potentials. Ann. Probab. 36(4), 1528–1583 (2008)

  4. 4

    Sznitman A.-S.: Brownian motion, obstacles and random media. Springer Monographs in Mathematics. Springer, Berlin (1998)

  5. 5

    Wang W.-M.: Mean field bounds on Lyapunov exponents in \({\mathbb {Z}^d}\) at the critical energy. Probab. Theory Relat. Fields 119(4), 453–474 (2001)

  6. 6

    Wang W.-M.: Mean field upper and lower bounds on Lyapunov exponents. Am. J. Math. 124(5), 851–878 (2002)

  7. 7

    Zerner M.P.W.: Directional decay of the Green’s function for a random nonnegative potential on \({\mathbb {Z}^d}\). Ann. Appl. Probab. 8, 246–280 (1998)

  8. 8

    Zerner M.P.W.: Velocity and Lyapounov exponents of some random walks in random environment. Ann. de l’IHP, Prob. et Stat 36(6), 737–748 (2000)

  9. 9

    Zygouras N.: Lyapounov norms for random walks in low disorder and dimension greater than three. Probab. Theory Relat. Fields 143(3–4), 615–642 (2009)

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Correspondence to Martin P. W. Zerner.

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Kosygina, E., Mountford, T.S. & Zerner, M.P.W. Lyapunov exponents of Green’s functions for random potentials tending to zero. Probab. Theory Relat. Fields 150, 43–59 (2011).

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  • Annealed
  • Green’s function
  • Lyapunov exponent
  • Quenched
  • Random potential
  • Random walk

Mathematics Subject Classification (2000)

  • 60K37
  • 82B41
  • 82B44