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Lyapunov exponents of Green’s functions for random potentials tending to zero
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  • Published: 11 February 2010

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

  • Elena Kosygina1,
  • Thomas S. Mountford2 &
  • Martin P. W. Zerner3 

Probability Theory and Related Fields volume 150, pages 43–59 (2011)Cite this article

  • 153 Accesses

  • 11 Citations

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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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Authors and Affiliations

  1. Department of Mathematics, Baruch College, Box B6-230, One Bernard Baruch Way, New York, NY, 10010, USA

    Elena Kosygina

  2. Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, 1015, Lausanne, Switzerland

    Thomas S. Mountford

  3. Mathematisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076, Tübingen, Germany

    Martin P. W. Zerner

Authors
  1. Elena Kosygina
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  2. Thomas S. Mountford
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  3. Martin P. W. Zerner
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Corresponding author

Correspondence to Martin P. W. Zerner.

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Cite this article

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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  • Received: 26 March 2009

  • Revised: 16 January 2010

  • Published: 11 February 2010

  • Issue Date: June 2011

  • DOI: 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
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