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

Abstract

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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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). https://doi.org/10.1007/s00440-010-0266-y

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Keywords

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

Mathematics Subject Classification (2000)

  • 60K37
  • 82B41
  • 82B44