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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- 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