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Lyapunov Exponents of Random Walks in Small Random Potential: The Lower Bound

Mountford, Thomas ; Mourrat, Jean-Christophe

In: Communications in Mathematical Physics, 2013, vol. 323, no. 3, p. 1071-1120

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    Titre
    • Lyapunov Exponents of Random Walks in Small Random Potential: The Lower Bound
    Auteur
    • Mountford, Thomas. Ecole Polytechnique Fédérale de Lausanne, Institut de Mathématiques, Station 8, 1015, Lausanne, Switzerland
    • Mourrat, Jean-Christophe. Ecole Polytechnique Fédérale de Lausanne, Institut de Mathématiques, Station 8, 1015, Lausanne, Switzerland
    Type de document
    • Postprint
    Langue
    • Inglese
    Publié dans
    • Communications in Mathematical Physics, 2013, vol. 323, no. 3, p. 1071-1120. Springer Berlin Heidelberg
    Autre version électronique
    Classification
    • Mathématiques
    Identifiant OAI-PMH
    • oai:doc.rero.ch:312268
    Summary
    We consider the simple random walk on $${\mathbb{Z}^d}$$ Z d , d > 3, evolving in a potential of the form β V, where $${(V(x))_{x \in \mathbb{Z}^d}}$$ ( V ( x ) ) x ∈ Z d are i.i.d. random variables taking values in [0,+∞), and β > 0. When the potential is integrable, the asymptotic behaviours as β tends to 0 of the associated quenched and annealed Lyapunov exponents are known (and coincide). Here, we do not assume such integrability, and prove a sharp lower bound on the annealed Lyapunov exponent for small β. The result can be rephrased in terms of the decay of the averaged Green function of the Anderson Hamiltonian $${-\triangle + \beta V}$$ - ▵ + β V